arxiv: 2605.00037 · v1 · submitted 2026-04-28 · 🌀 gr-qc · hep-th

Recognition: unknown

Topology of black hole thermodynamics: A brief review

Shao-Wen Wei , Yu-Xiao Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:59 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black hole thermodynamicstopological numbersuniversality classesphase transitionsHawking-Page transitionDavies pointscritical pointsthermodynamic limit

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The pith

Topological numbers derived from black hole thermodynamic potentials place different systems into distinct universality classes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews recent work showing that topological methods applied to black hole thermodynamics produce numbers that remain the same for whole classes of black holes. These numbers become especially clear when systems are taken to the thermodynamic limit and help organize phase transitions and critical points. The review walks through the basic topological setup for black hole solutions, critical points, Davies points, and the Hawking-Page transition, then computes the numbers and explains what they mean for each case.

Core claim

By extracting topological numbers from the thermodynamic potentials of black holes, systems can be grouped into universality classes; the grouping holds most cleanly in the thermodynamic limit and supplies a route toward a broader quantum gravity description.

What carries the argument

Topological numbers calculated from thermodynamic potentials, which label phase structures and remain unchanged across different black hole solutions.

If this is right

Black hole systems that share the same topological number belong to the same universality class in their thermodynamics.
The classification becomes sharper when the black holes are examined in the thermodynamic limit.
Phase transitions such as the Hawking-Page transition and critical points receive a consistent topological label.
Davies points and other critical behaviors acquire topological invariants that distinguish or group them.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the classes prove stable, they could serve as a coarse-grained way to compare thermodynamic properties across very different spacetimes without solving the full equations each time.
The same topological counting might eventually be applied to rotating or charged black holes in higher dimensions to see whether new classes appear.
A mismatch between the topological label and observed phase behavior in a concrete black hole would indicate where the thermodynamic approximation breaks down.

Load-bearing premise

The topological numbers stay the same and keep their physical meaning no matter which coordinates or approximations are chosen for a given black hole solution.

What would settle it

A calculation for one black hole where the topological number shifts after a coordinate redefinition or after switching to a different approximation for the same solution would show the classification does not hold.

Figures

Figures reproduced from arXiv: 2605.00037 by Shao-Wen Wei, Yu-Xiao Liu.

**Figure 2.** Figure 2: FIG. 2: Ω vs [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: The red arrows represent the vector field [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Ω vs [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: (a) The blue arrows represent the vector field [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: The red arrows represent the unit vector field [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Zero points of the vector [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Zero points of [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

read the original abstract

Recent explorations of topological aspects in black hole thermodynamics have achieved unprecedented progress. By utilizing topological numbers, different black hole systems can be categorized into distinct universality classes. This universal classification is particularly evident in thermodynamic limits, offering valuable insights for developing a comprehensive quantum gravity framework. This review highlights the latest advancements in this field. Specifically, we outline fundamental topological frameworks underlying black hole solutions, critical points, Davies points, and the Hawking-Page phase transition. For each scenario, we calculate the associated topological numbers and analyze their physical significance. Furthermore, we explore the practical implications arising from this research.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a review that collects existing calculations of topological numbers for black hole thermodynamics but adds no new derivations or invariance checks.

read the letter

This review gathers prior work on topological numbers in black hole thermodynamics and walks through the standard calculations for critical points, Davies points, and the Hawking-Page transition. It shows how those numbers are used to sort different black hole systems into classes and notes possible links to quantum gravity ideas. The presentation is organized and sticks to the cited frameworks without introducing fresh math or data of its own. That makes it a convenient summary for anyone who wants the main results in one place rather than hunting through the original papers. The calculations themselves follow the usual thermodynamic potential approach and report the winding numbers or charges as previously derived. The main limitation is that the universality claim rests on those numbers being stable across coordinate choices, gauges, and ensembles. The review reports the numbers from the literature but does not include any explicit test showing they remain unchanged when the same physical system is rewritten in different coordinates or thermodynamic variables. If that stability fails, the classification loses much of its claimed robustness. Readers already active in this corner of black hole physics will likely find the summary useful for quick reference but will not gain new results to build on. Someone entering the area could use it to see the current state of the calculations. The paper is coherent on its own terms and engages the existing literature directly. It deserves peer review if a journal wants a concise overview of this specific line of work, though it will not stand as an original contribution.

Referee Report

1 major / 2 minor

Summary. This paper is a brief review of recent work applying topological methods to black hole thermodynamics. It outlines fundamental topological frameworks for black hole solutions, critical points, Davies points, and the Hawking-Page phase transition. For each case the review calculates the associated topological numbers, analyzes their physical significance, and argues that these numbers allow different black hole systems to be grouped into distinct universality classes, with particular emphasis on thermodynamic limits and implications for quantum gravity.

Significance. If the topological numbers are robust, the review consolidates an emerging approach that could classify black-hole thermodynamics in a model-independent way and supply new constraints for quantum gravity. As a review it is useful for mapping the literature, but its long-term significance hinges on whether the claimed universality survives coordinate and ensemble changes.

major comments (1)

[Outline of topological frameworks for critical points and Hawking-Page transition] The central claim that topological numbers define coordinate-independent universality classes is load-bearing for the entire review. In the sections outlining the topological frameworks for critical points and the Hawking-Page transition, the calculations are presented for specific metric ansätze and thermodynamic potentials, yet no explicit check or citation is given showing that the winding number or topological charge is unchanged when the identical physical system is rewritten in a different coordinate chart or gauge. Without such invariance the universality classes lose physical meaning.

minor comments (2)

The term 'thermodynamic limits' is used in the abstract and introduction but is not defined or illustrated with a concrete example; a short clarifying paragraph would improve readability.
Ensure that every cited derivation is accompanied by a precise reference to the original equation or section in the source paper so readers can verify the reproduced topological numbers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our review. We address the major concern point by point below.

read point-by-point responses

Referee: [Outline of topological frameworks for critical points and Hawking-Page transition] The central claim that topological numbers define coordinate-independent universality classes is load-bearing for the entire review. In the sections outlining the topological frameworks for critical points and the Hawking-Page transition, the calculations are presented for specific metric ansätze and thermodynamic potentials, yet no explicit check or citation is given showing that the winding number or topological charge is unchanged when the identical physical system is rewritten in a different coordinate chart or gauge. Without such invariance the universality classes lose physical meaning.

Authors: We agree that explicit demonstration of coordinate and gauge invariance is essential to support the physical interpretation of the universality classes. The topological charge is defined via a vector field constructed from thermodynamic potentials (temperature, entropy, pressure, etc.), which are scalar quantities independent of coordinate choice. Nevertheless, the review does not contain an explicit invariance check or dedicated citation for this property in the critical-point and Hawking-Page sections. We will therefore add a short dedicated paragraph (with supporting references from the existing literature) that outlines why the winding number remains unchanged under reparametrization of the metric ansatz and under Legendre transforms between ensembles. This addition will be placed immediately after the outline of the topological frameworks. revision: yes

Circularity Check

0 steps flagged

Review summarizes external derivations; no internal circularity

full rationale

This paper is explicitly a review that outlines fundamental topological frameworks, calculates associated topological numbers for scenarios like critical points and Hawking-Page transitions, and analyzes their significance by referencing prior literature. No new derivations, parameter fits, or self-contained proofs are presented that could reduce to inputs by construction. The universality classification claim is framed as evident from the summarized external results rather than derived here. Absent any quoted equations or steps within the manuscript that equate a prediction to a fitted input or self-citation chain, the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This review summarizes prior literature and does not introduce new free parameters, axioms, or invented entities beyond those already present in the referenced topological thermodynamics papers.

pith-pipeline@v0.9.0 · 5383 in / 932 out tokens · 31156 ms · 2026-05-09T20:59:20.354422+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

211 extracted references · 162 canonical work pages · 2 internal anchors

[1]

This example is taken from a conversation between John Wheeler and Jakob Bekenstein
[2]

Bekenstein,Black holes and the second law, Lett

J. Bekenstein,Black holes and the second law, Lett. Nuovo Cim.4, 737 (1972)

1972
[3]

J. D. Bekenstein,Black holes and entropy, Phys. Rev. D7, 2333 (1973)

1973
[4]

S. W. Hawking,Black hole explosions, Nature248, 30 (1974)

1974
[5]

S. W. Hawking,Particle creation by black holes, Com- mun. Math. Phys.43, 199 (1975)

1975
[6]

J. M. Bardeen, B. Carter, and S. Hawking,The four laws of black hole mechanics, Commun. Math. Phys. 16 31, 161 (1973)

1973
[7]

S. W. Hawking, and D. N. Page,Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys.87, 577 (1983)

1983
[8]

Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories

E. Witten,Anti-de Sitter space, thermal phase transi- tion, and confinement in gauge theories, Adv. Theor. Math. Phys.2, 505 (1998), [arXiv:hep-th/9803131]

work page Pith review arXiv 1998
[9]

J. M. Maldacena,The largeNlimit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998), [arXiv:hep-th/9711200]

work page internal anchor Pith review arXiv 1998
[10]

S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B428, 105 (1998), [arXiv:hep-th/9802109]

work page internal anchor Pith review arXiv 1998
[11]

Anti de Sitter space and holography,

E. Witten,Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2, 253 (1998), [arXiv:hep- th/9802150]

work page arXiv 1998
[12]

Enthalpy and the Mechanics of AdS Black Holes

D. Kastor, S. Ray, and J. Traschen,Enthalpy and the mechanics of AdS black holes, Class. Quant. Grav.26, 195011 (2009), [arXiv:0904.2765 [hep-th]]

work page Pith review arXiv 2009
[13]

P-V criticality of charged AdS black holes

D. Kubiznak, and R. B. Mann,P-Vcriticality of charged AdS black holes, J. High Energy Phys.1207, 033 (2012), [arXiv:1205.0559 [hep-th]]

work page Pith review arXiv 2012
[14]

Altamirano, D

N. Altamirano, D. Kubiznak, and R. B. Mann,Reen- trant phase transitions in rotating AdS black holes, Phys. Rev. D88, 101502(R) (2013), [arXiv:1306.5756 [hep-th]]

work page arXiv 2013
[15]

Kerr-AdS analogue of triple point and solid/liquid/gas phase transition,

N. Altamirano, D. Kubiznak, R. B. Mann, and Z. Sherkatghanad,Kerr-AdS analogue of triple point and solid/liquid/gas phase transition, Class. Quant. Grav. 31, 042001 (2014), [arXiv:1308.2672 [hep-th]]

work page arXiv 2014
[16]

Altamirano, D

N. Altamirano, D. Kubiznak, R. B. Mann, and Z. Sherkatghanad,Thermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume, Galaxies2, 89 (2014), [arXiv:1401.2586 [hep- th]]

work page arXiv 2014
[17]

B. P. Dolan, A. Kostouki, D. Kubiznak, and R. B. Mann,Isolated critical point from Lovelock gravity, Class. Quant. Grav.31, 242001 (2014), [arXiv:1407.4783 [hep-th]]

work page arXiv 2014
[18]

Wei, and Y.-X

S.-W. Wei, and Y.-X. Liu,Critical phenomena and thermodynamic geometry of charged Gauss-Bonnet AdS black holes, Phys. Rev. D87, 044014 (2013), [arXiv:1209.1707 [gr-qc]]

work page arXiv 2013
[19]

Triple points and phase diagrams in the extended phase space of charged Gauss-Bonnet black holes in AdS space,

S.-W. Wei, and Y.-X. Liu,Triple points and phase dia- grams in the extended phase space of charged Gauss- Bonnet black holes in AdS space, Phys. Rev. D90, 044057 (2014), [arXiv:1402.2837 [hep-th]]

work page arXiv 2014
[20]

A. M. Frassino, D. Kubiznak, R. B. Mann, and F. Simovic,Multiple reentrant phase transitions and triple points in Lovelock thermodynamics, J. High Energy Phys.1409, 080 (2014), [arXiv:1406.7015 [hep-th]]

work page arXiv 2014
[21]

P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space

R.-G. Cai, L.-M. Cao, L. Li, and R.-Q. Yang,P-Vcriti- cality in the extended phase space of Gauss-Bonnet black holes in AdS space, J. High Energy Phys.1309, 005 (2013), [arXiv:1306.6233 [gr-qc]]

work page Pith review arXiv 2013
[22]

H. Xu, W. Xu, and L. Zhao,Extended phase space thermodynamics for third order Lovelock black holes in diverse dimensions, Eur. Phys. J. C74, 3074 (2014), [arXiv:1405.4143 [gr-qc]]

work page arXiv 2014
[23]

R. A. Hennigar, W. G. Brenna, and R. B. Mann,P- Vcriticality in quasitopological gravity, J. High Energy Phys.1507, 077 (2015), [arXiv:1505.05517 [hep-th]]

work page arXiv 2015
[24]

R. A. Hennigar, and R. B. Mann,Reentrant phase tran- sitions and van der Waals behaviour for hairy black holes, Entropy17, 8056 (2015), [arXiv:1509.06798 [hep- th]]

work page arXiv 2015
[25]

R. A. Hennigar, R. B. Mann, and E. Tjoa,Super- fluid black holes, Phys. Rev. Lett.118, 021301 (2017), [arXiv:1609.02564 [hep-th]]

work page arXiv 2017
[26]

Black hole chem- istry: thermodynamics with Lambda,

D. Kubiznak, R. B. Mann, and M. Teo,Black hole chem- istry: thermodynamics with Lambda, Class. Quantum Grav.34, 063001 (2017), [arXiv:1608.06147 [hep-th]]

work page arXiv 2017
[27]

Z.-M. Xu, B. Wu, and W.-L. Yang,Rate of the phase transition for a charged anti-de Sitter black hole, Sci. China Phys. Mech. Astron.66, 240411 (2023), [arXiv:2211.03512 [gr-qc]]

work page arXiv 2023
[28]

Yang, S.-P

S.-J. Yang, S.-P. Wu, S.-W. Wei, and Y.-X. Liu,De- ciphering black hole phase transitions through photon spheres, Sci. China Phys. Mech. Astron.68, 120412 (2025), [arXiv:2505.20860 [gr-qc]]

work page arXiv 2025
[29]

Ma, H.-F

M.-S. Ma, H.-F. Li, and J.-H. Shi,Regular black holes and reductions of thermodynamic phase spaces, Sci. China Phys. Mech. Astron.69, 210411 (2026), [arXiv:2507.09551 [gr-qc]]

work page arXiv 2026
[30]

R. B. Mann,Black hole chemistry: The first 15 years, Int. J. Mod. Phys. D34, 2542001 (2025), [arXiv:2508.01830 [gr-qc]]

work page arXiv 2025
[31]

P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro,Light ring stability in ultra-compact objects, Phys. Rev. Lett. 119, 251102 (2017), [arXiv:1708.04211 [gr-qc]]

work page arXiv 2017
[32]

P. V. P. Cunha, and C. A. R. Herdeiro,Stationary black holes and light rings, Phys. Rev. Lett.124, 181101 (2020), [arXiv:2003.06445 [gr-qc]]

work page arXiv 2020
[33]

Wei, Topological Charge and Black Hole Photon Spheres, Phys

S.-W. Wei,Topological Charge and Black Hole Pho- ton Spheres, Phys. Rev. D102, 064039 (2020), [arXiv:2006.02112 [gr-qc]]

work page arXiv 2020
[34]

Wei, and Y.-X

S.-W. Wei, and Y.-X. Liu,Topology of equatorial time- like circular orbits around stationary black holes, Phys. 17 Rev. D107, 064006 (2023), [arXiv:2207.08397 [gr-qc]]

work page arXiv 2023
[35]

Duan, and M.-L

Y.-S. Duan, and M.-L. Ge,SU(2) gauge theory and electrodynamics ofNmoving magnetic monopoles, Sci. Sin.9, 1072 (1979)

1979
[36]

Duan,The structure of the topological current, SLAC-PUB-3301, (1984)

Y.-S. Duan,The structure of the topological current, SLAC-PUB-3301, (1984)

1984
[37]

Fu, Y.-S

L.-B. Fu, Y.-S. Duan, and H. Zhang,Evolution of the Chern-Simons vortices, Phys. Rev. D61, 045004 (2000), [arXiv:hep-th/0112033]

work page arXiv 2000
[38]

Wei and Y.-X

S.-W. Wei, and Y.-X. Liu,Topology of black hole thermodynamics, Phys. Rev. D105, 104003 (2022), [arXiv:2112.01706 [gr-qc]]

work page arXiv 2022
[39]

S.-W. Wei, P. Cheng, and Y.-X. Liu,Analytical and ex- act critical phenomena of -dimensional singly spinning Kerr-AdS black holes, Phys. Rev. D93, 084015 (2016), [arXiv:1510.00085 [gr-qc]]

work page arXiv 2016
[40]

Fernando, and D

S. Fernando, and D. Krug,Charged black hole solu- tions in Einstein-Born-Infeld gravity with a cosmological constant, Gen. Rel. Grav.35, 129 (2003), [arXiv:hep- th/0306120]

work page arXiv 2003
[41]

Gunasekaran, R

S. Gunasekaran, D. Kubiznak, and R. B. Mann,Ex- tended phase space thermodynamics for charged and ro- tating black holes and Born-Infeld vacuum polarization, J. High Energy Phys.11, 110 (2012), [arXiv:1208.6251 [hep-th]]

work page arXiv 2012
[42]

P. K. Yerra, and C. Bhamidipati,Topology of black hole thermodynamics in Gauss-Bonnet gravity, Phys. Rev. D 105, 104053 (2022), [arXiv:2202.10288 [gr-qc]]

work page arXiv 2022
[43]

M. B. Ahmed, D. Kubiznak, and R. B. Mann, Vortex/anti-vortex pair creation in black hole ther- modynamics, Phys. Rev. D107, 046013 (2023), [arXiv:2207.02147 [hep-th]]

work page arXiv 2023
[44]

P. K. Yerra, and C. Bhamidipati,Topology of Born- Infeld AdS black holes in 4D novel Einstein-Gauss- Bonnet gravity, Phys. Lett. B835, 137591 (2022), [arXiv:2207.10612 [gr-qc]]

work page arXiv 2022
[45]

P. K. Yerra, C. Bhamidipati, and S. Mukherji,Topology of critical points in boundary matrix duals, J. High En- ergy Phys.03, 138 (2024), [arXiv:2304.14988 [hep-th]]

work page arXiv 2024
[46]

Zhang, H.-Y

M.-Y. Zhang, H.-Y. Zhou, H. Chen, H. Hassanabadi, and Z.-W. Long,Topological classification of critical points for hairy black holes in Lovelock gravity, Eur. Phys. J. C84, 1251 (2024)

2024
[47]

P. C. W. Davies,The thermodynamic theory of black holes, Proc. Roy. Soc. Lond. A353, 499 (1977)

1977
[48]

Bhattacharya, K

K. Bhattacharya, K. Bamba, and D. Singleton,Topo- logical interpretation of extremal and Davies-type phase transitions of black holes, Phys. Lett. B854, 138722 (2024) [arXiv:2402.18791 [gr-qc]]

work page arXiv 2024
[49]

Hazarika, N

B. Hazarika, N. J. Gogoi, and P. Phukon,Revis- iting thermodynamic topology of Hawking-Page and Davies type phase transitions, JHEAp45, 87 (2025), [arXiv:2404.02526 [hep-th]]

work page arXiv 2025
[50]

Mehmood, N

A. Mehmood, N. Alessa, M. U. Shahzad, and E. E. Zo- tos,Davies-type phase transitions in 4D Dyonic AdS black holes from topological perspective, Nucl. Phys. B 1006, 116653 (2024)

2024
[51]

P. K. Yerra, C. Bhamidipati, and S. Mukherji,Topology of critical points and Hawking-Page transition, Phys. Rev. D106, 064059 (2022), [arXiv:2208.06388 [hep-th]]

work page arXiv 2022
[52]

P. K. Yerra, C. Bhamidipati, and S. Mukherji,Topol- ogy of Hawking-Page transition in Born-Infeld AdS black holes, J. Phys. Conf. Ser.2667, 012031 (2023), [arXiv:2312.10784 [gr-qc]]

work page arXiv 2023
[53]

Black Hole So- lutions as Topological Thermodynamic Defects,

S.-W. Wei, Y.-X. Liu, and R. B. Mann,Black Hole So- lutions as Topological Thermodynamic Defects, Phys. Rev. Lett.129, 191101 (2022), [arXiv:2208.01932 [gr- qc]]

work page arXiv 2022
[54]

J. W. York,Black hole thermodynamics and the Eu- clidean Einstein action, Phys. Rev. D33, 2092 (1986)

2092
[55]

Wu, S.-J

S.-P. Wu, S.-J. Yang, and S.-W. Wei,Extended thermo- dynamical topology of black hole, Eur. Phys. J. C85, 1372 (2025), [arXiv:2508.01614 [hep-th]]

work page arXiv 2025
[56]

Wu,Topological classes of rotating black holes, Phys

D. Wu,Topological classes of rotating black holes, Phys. Rev. D107, 024024 (2023), [arXiv:2211.15151 [gr-qc]]

work page arXiv 2023
[57]

A. S. Mohamed, M. U. Shahzad, A. Mehmood, and M. Z. Ashraf,Thermodynamic topology of rotating regular black hole in conformal massive gravity in CMG, Int. J. Mod. Phys. A40, 2550022 (2025)

2025
[58]

Wu, and S.-Q

D. Wu, and S.-Q. Wu,Topological classes of thermody- namics of rotating AdS black holes, Phys. Rev. D107, 084002 (2023), [arXiv:2301.03002 [hep-th]]

work page arXiv 2023
[59]

X.-D. Zhu, D. Wu, and D. Wen,Topological classes of thermodynamics of the rotating charged AdS black holes in gauged supergravities, Phys. Lett. B856, 138919 (2024), [arXiv:2402.15531 [hep-th]]

work page arXiv 2024
[60]

A. H. Taub,Empty space-times admitting a three pa- rameter group of motions, Ann. Math.53, 472 (1951)

1951
[61]

Newman, L.Tamburino, and T

E. Newman, L.Tamburino, and T. Unti,Empty-space generalization of the schwarzschild metric, J. Math. Phys.4, 915 (1963)

1963
[62]

Kinnersley, and M

W. Kinnersley, and M. Walker,Uniformly accelerating charged mass in general relativity, Phys. Rev. D2, 1359 (1970)

1970
[63]

Wu,Topological classes of thermodynamics of the four-dimensional static accelerating black holes, Phys

D. Wu,Topological classes of thermodynamics of the four-dimensional static accelerating black holes, Phys. Rev. D108, 084041 (2023), [arXiv:2307.02030 [hep-th]]

work page arXiv 2023
[64]

W. Liu, L. Zhang, D. Wu, and J. Wang,Thermody- 18 namic topological classes of the rotating, accelerating black holes, Class. Quant. Grav.42, 125007 (2025), [arXiv:2409.11666 [hep-th]]

work page arXiv 2025
[65]

Wu,Classifying topology of consistent thermodynam- ics of the four-dimensional neutral Lorentzian NUT- charged spacetimes, Eur

D. Wu,Classifying topology of consistent thermodynam- ics of the four-dimensional neutral Lorentzian NUT- charged spacetimes, Eur. Phys. J. C83, 365 (2023), [arXiv:2302.01100 [gr-qc]]

work page arXiv 2023
[66]

Wu,Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes, Eur

D. Wu,Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes, Eur. Phys. J. C83, 589 (2023), [arXiv:2306.02324 [gr-qc]]

work page arXiv 2023
[67]

Azreg-A¨ ınou, and M

M. Azreg-A¨ ınou, and M. Rizwan,Thermodynamic topo- logical classes of family of black holes with NUT-charge, JCAP05, 091 (2025), [arXiv:2502.13914 [gr-qc]]

work page arXiv 2025
[68]

M. U. Shahzad, N. Alessa, A. Mehmood, and R. Javed, Thermodynamic topology of Hot NUT-Kerr-Newman- Kasuya-Anti-de Sitter black hole, Astron. Comput.51, 100900 (2025)

2025
[69]

Liu, and J

C. Liu, and J. Wang,Topological natures of the Gauss- Bonnet black hole in AdS space, Phys. Rev. D107, 064023 (2023), [arXiv:2211.05524 [gr-qc]]

work page arXiv 2023
[70]

N.-C. Bai, L. Song, and J. Tao,Reentrant phase transition in holographic thermodynamicsof Born–Infeld AdS black hole, Eur. Phys. J. C84, 43 (2024), [arXiv:2212.04341 [hep-th]]

work page arXiv 2024
[71]

R. Li, C. Liu, K. Zhang, and J. Wang,Topology of the landscape and dominant kinetic path for the thermody- namic phase transition of the charged Gauss-Bonnet- AdS black holes, Phys. Rev. D108, 044003 (2023), [arXiv:2302.06201 [gr-qc]]

work page arXiv 2023
[72]

Li, and J

R. Li, and J. Wang,Generalized free energy land- scapes of charged Gauss-Bonnet-AdS black holes in di- verse dimensions, Phys. Rev. D108, 044057 (2023), [arXiv:2304.03425 [gr-qc]]

work page arXiv 2023
[73]

M. S. Ali, H. El Moumni, J. Khalloufi, and K. Masmar, Topology of Born–Infeld-AdS black hole phase transi- tions: Bulk and CFT sides, Annals Phys.465, 169679 (2024), [arXiv:2306.11212 [hep-th]]

work page arXiv 2024
[74]

Du, H.-F

Y.-Z. Du, H.-F. Li, Y.-B. Ma, and Q. Gu,Topol- ogy and phase transition for EPYM AdS black hole in thermal potential, Nucl. Phys. B1006, 116641 (2024), [arXiv:2309.00224 [hep-th]]

work page arXiv 2024
[75]

M. U. Shahzad, A. Mehmood, S. Sharif, and A. ¨Ovg¨ un, Criticality and topological classes of neutral Gauss- Bonnet AdS black holes in 5D, Annals Phys.458, 169486 (2023)

2023
[76]

M. U. Shahzad, A. Mehmood, R. Gohar, and A. ¨Ovg¨ un, Joule Thomson expansion, Maxwell equal area law and topological interpretation of Phantom RN AdS black holes, New Astron.110, 102225 (2024)

2024
[77]

Chen, M.-Y

H. Chen, M.-Y. Zhang, H. Hassanabadi, B. C. L¨ utf¨ uo˘ glu, and Z.-W. Long,Thermodynamic topol- ogy of dyonic AdS black holes with quasitopologi- cal electromagnetism in Einstein-Gauss-Bonnet gravity, [arXiv:2403.14730 [gr-qc]]

work page arXiv
[78]

H. Chen, D. Wu, M.-Y. Zhang, H. Hassanabadi, and Z.-W. Long,Thermodynamic topology of phantom AdS black holes in massive gravity, Phys. Dark Univ.46, 101617 (2024), [arXiv:2404.08243 [gr-qc]]

work page arXiv 2024
[79]

Hazarika, and P

B. Hazarika, and P. Phukon,Topology of restricted phase space thermodynamics in Kerr-Sen-Ads black holes, Nucl. Phys. B1012, 116837 (2025), [arXiv:2405.02328 [hep-th]]

work page arXiv 2025
[80]

Wang, and Y.-Z

H. Wang, and Y.-Z. Du,Topology of charged AdS black hole in restricted phase space, Chin. Phys. C48, 095109 (2024), [arXiv:2406.08793 [hep-th]]

work page arXiv 2024

Showing first 80 references.