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Black Hole Entropy Beyond the Wald Term in Nonminimally Coupled Gravity: A Covariant Phase Space Decomposition

T0 review · 2 major / 1 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read In nonminimally coupled gravity, black hole entropy includes terms beyond the Wald entropy when matter fields cannot extend smoothly to the bifurcation surface.

desk verdict The paper decomposes black hole entropy beyond Wald for nonminimal couplings but the split may not be independent of surface charge representative. read the letter →

arxiv 2605.22429 v1 pith:F6TJG4XF submitted 2026-05-21 gr-qc hep-th

Jia-Zhou Liu , Shan-Ping Wu , Shao-Wen Wei , Yu-Xiao Liu This is my paper · ORCID
classification gr-qchep-th
keywords blackholeentropyWaldnonminimalcouplingcovariantphasespaceKalb-RamondfieldbumblebeegravityGauss-Bonnetdiffeomorphisminvariance
checked against Cost.FunctionalEquationabsolute_floor_iff_bare_distinguishability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that for static spherically symmetric black holes in diffeomorphism-invariant theories with nonminimal matter-curvature couplings, the entropy that enters the first law is not limited to the standard Wald entropy. Instead, when a matter field cannot be smoothly extended to the regular bifurcation surface, the horizon surface charge variation yields extra finite contributions that must be included. A sympathetic reader would care because this ensures thermodynamic consistency in a wider range of modified gravity models and identifies precisely when the Wald formula alone is insufficient. The decomposition is obtained by applying the covariant phase space formalism and subtracting ordinary work terms.

What carries the argument

The decomposition of the horizon surface charge variation in the covariant phase space formalism, after subtracting work terms, into the Wald entropy plus the non-Wald Noether contribution S_1 and the remaining integrable term ΔS.

What would settle it

Perform an independent calculation of the entropy for a bumblebee black hole solution, for example via the Euclidean action or by direct integration of the first law, and check whether the result equals only the Wald term or requires the additional S_1 and ΔS contributions.

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Extended reading notes

Core claim

For regular bifurcate Killing horizons the Iyer-Wald construction recovers the Wald entropy, but when matter fields fail to extend smoothly to the bifurcation surface the horizon surface charge variation contains additional finite pieces. After ordinary work terms are subtracted, the entropy entering the first law decomposes as S_H = S_W + S_1 + ΔS, where S_W is the Wald entropy, S_1 is the non-Wald part of the Noether charge, and ΔS is the remaining integrable part of the surface charge variation.

Load-bearing premise

The horizon is a regular bifurcate Killing horizon and the covariant phase space formalism applies without further restrictions on the matter-field extension or the choice of representative for the surface charge variation.

Editorial extensions

If this is right

  • For the regular Kalb-Ramond branch the entropy reduces exactly to the Wald term.
  • Bumblebee branches produce either a nonzero ΔS with vanishing S_1 or a cancellation between S_1 and ΔS.
  • Weyl-vector extended Gauss-Bonnet examples require nonzero contributions from both S_1 and ΔS.
  • The criterion directly shows whether the Wald entropy density alone satisfies the first law or whether the full surface charge variation is needed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition can be applied to other diffeomorphism-invariant theories with nonminimal couplings to test whether extra terms appear.
  • The results suggest that the smoothness of matter-field extension near the horizon controls whether thermodynamic relations receive corrections beyond the Wald formula.
  • Independent entropy computations for these specific solutions would confirm whether the decomposed expression or the pure Wald term matches other methods.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper applies the covariant phase space formalism to static spherically symmetric black holes in diffeomorphism-invariant theories with nonminimal matter-curvature couplings. For regular bifurcate Killing horizons, it decomposes the entropy entering the first law as S_H = S_W + S_1 + ΔS when matter fields cannot be smoothly extended to the bifurcation surface, with S_W the Wald entropy, S_1 the non-Wald part of the Noether charge, and ΔS the remaining integrable contribution after subtracting ordinary work terms. The decomposition is evaluated on Kalb-Ramond, bumblebee, and extended Gauss-Bonnet examples, yielding S_H = S_W for the regular Kalb-Ramond branch, either S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee branches, and both corrections required for the Weyl-vector extended Gauss-Bonnet cases.

Significance. If the decomposition is robust, the work supplies a practical criterion for deciding when the Wald entropy density suffices and when the full horizon surface charge variation must be retained. The concrete results for three model classes furnish falsifiable predictions and illustrate how non-smooth matter extensions generate additional integrable contributions. This strengthens the covariant phase space approach for nonminimally coupled theories.

major comments (2)
  1. [Abstract] Abstract (paragraph beginning 'In the representative obtained by directly varying the action'): the decomposition is performed in one specific representative of the surface charge. The covariant phase space formalism permits addition of exact forms to the Noether current without changing the equations of motion. No demonstration is given that the split S_H = S_W + S_1 + ΔS or the total S_H remains unchanged under such additions. Because the central claim concerns the necessity of terms beyond S_W, invariance under representative choice is load-bearing and should be shown explicitly.
  2. [Applications to models] Applications section (Kalb-Ramond, bumblebee, and extended Gauss-Bonnet examples): the reported outcomes (S_H = S_W for regular Kalb-Ramond; S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee; both corrections for Weyl-vector Gauss-Bonnet) rest on the chosen representative and on the assumed regularity of the bifurcate horizon. Explicit expressions for the surface charge variations and the subtracted work terms would allow verification that the reported patterns are not artifacts of the representative choice.
minor comments (1)
  1. [Notation] The abstract introduces S_1 and ΔS without an equation number; a numbered display of the decomposition S_H = S_W + S_1 + ΔS in the main text would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments correctly identify that the manuscript presents the entropy decomposition in a specific representative of the Noether current and that the applications would benefit from more explicit intermediate expressions. We address both points below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'In the representative obtained by directly varying the action'): the decomposition is performed in one specific representative of the surface charge. The covariant phase space formalism permits addition of exact forms to the Noether current without changing the equations of motion. No demonstration is given that the split S_H = S_W + S_1 + ΔS or the total S_H remains unchanged under such additions. Because the central claim concerns the necessity of terms beyond S_W, invariance under representative choice is load-bearing and should be shown explicitly.

    Authors: We agree that an explicit demonstration of invariance is needed. In the revised manuscript we will add a short subsection (in the general formalism section) showing that the addition of an exact form dα to the Noether current does not change the decomposition or the total integrable entropy S_H. On a closed bifurcation surface the integral of the exact term vanishes identically, and any residual boundary contributions cancel against the subtracted work terms when the first law is assembled. This establishes that the split S_H = S_W + S_1 + ΔS is representative-independent for the class of theories and horizons considered. revision: yes

  2. Referee: [Applications to models] Applications section (Kalb-Ramond, bumblebee, and extended Gauss-Bonnet examples): the reported outcomes (S_H = S_W for regular Kalb-Ramond; S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee; both corrections for Weyl-vector Gauss-Bonnet) rest on the chosen representative and on the assumed regularity of the bifurcate horizon. Explicit expressions for the surface charge variations and the subtracted work terms would allow verification that the reported patterns are not artifacts of the representative choice.

    Authors: We accept that the applications section would be more transparent with the intermediate expressions. In the revision we will add an appendix containing the explicit forms of the horizon surface charge variation δQ_H and the subtracted work terms for each of the three models. These expressions will be derived from the same representative used in the main text, allowing direct verification that the reported results (S_H = S_W for the regular Kalb-Ramond branch, S_1 = 0 with ΔS ≠ 0 or cancellation for bumblebee branches, and both corrections for the Weyl-vector Gauss-Bonnet cases) follow from the general decomposition and are not artifacts of the representative choice. The regularity assumption for the bifurcate horizon is the standard one employed in the Iyer-Wald construction. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation applies standard formalism to specific representative

full rationale

The paper applies the covariant phase space formalism and Iyer-Wald procedure to decompose the horizon surface charge variation for regular bifurcate Killing horizons in nonminimally coupled theories. The split S_H = S_W + S_1 + ΔS is obtained after subtracting work terms from the representative chosen by direct action variation; this partitioning does not reduce any claimed result to a fitted parameter, self-defined quantity, or load-bearing self-citation by construction. The central claim is tested on concrete examples (Kalb-Ramond, bumblebee, extended Gauss-Bonnet) where the presence or absence of extra terms is exhibited explicitly. No equation equates the final entropy expression to its inputs tautologically, and the construction remains self-contained against external benchmarks in the literature.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The abstract invokes the standard covariant phase space formalism and the existence of regular bifurcate Killing horizons; no free parameters, ad-hoc axioms, or new postulated entities are introduced or fitted.

assumptions (2)
  • domain assumption The horizon is a regular bifurcate Killing horizon to which the Iyer-Wald construction applies.
    Stated in the opening sentence of the abstract as the setting in which the standard Wald entropy is recovered.
  • domain assumption The covariant phase space formalism yields a well-defined horizon surface charge variation after ordinary work terms are subtracted.
    Implicit in the decomposition S_H = S_W + S_1 + ΔS presented in the abstract.

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Pith. "Pith review of Black Hole Entropy Beyond the Wald Term in Nonminimally Coupled Gravity: A Covariant Phase Space Decomposition." pith.science (2026). https://pith.science/paper/F6TJG4XF

@misc{pith2026260522429,
  author       = {Pith},
  title        = {Pith review of: Black Hole Entropy Beyond the Wald Term in Nonminimally Coupled Gravity: A Covariant Phase Space Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F6TJG4XF}},
  note         = {Machine review of arXiv:2605.22429}
}
abstract

We study the entropy of static, spherically symmetric black holes in diffeomorphism-invariant theories with nonminimal matter--curvature couplings, using the covariant phase space formalism. For regular bifurcate Killing horizons, the Iyer--Wald construction gives the standard Wald entropy. If a matter field cannot be smoothly extended to the regular bifurcation surface, however, the horizon surface charge variation can contain finite contributions that are not included in the Wald entropy density. In the representative obtained by directly varying the action, and after ordinary work terms are subtracted, we decompose the entropy entering the first law of black hole thermodynamics as \(S_{\mathrm H}=S_{\mathrm W}+S_1+\Delta S\). Here \(S_{\mathrm W}\) is the Wald entropy, \(S_1\) is the non-Wald part of the Noether charge, and \(\Delta S\) is the remaining integrable part of the horizon surface charge variation. Applying this criterion to Kalb--Ramond, bumblebee, and extended Gauss--Bonnet black holes, we find that the regular Kalb--Ramond branch has \(S_{\mathrm H}=S_{\mathrm W}\), the bumblebee branches yield either \(S_1=0\) with \(\Delta S\neq0\) or a cancellation between \(S_1\) and \(\Delta S\), and the Weyl-vector extended Gauss--Bonnet examples require both corrections. This gives a direct test of whether the Wald entropy density is sufficient, or whether the full horizon surface charge variation has to be used.

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Forward citations

Cited by 2 Pith papers

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Works this paper leans on

58 extracted references · 58 canonical work pages · cited by 2 Pith papers

  1. [1]

    the family admits a Killing horizon generatorξ a = (∂t)a

    work page

  2. [2]

    (18) is integrable on the chosen solution family

    the entropy one form obtained from the equality be- tweenδH (∞) ξ and the horizon variation in Eq. (18) is integrable on the chosen solution family. Then SH =S W +S 1 + ∆S,(52) with the three terms defined in Eqs. (24), (26), and (33). For the restricted variation bδ, the remainders are bRX , bRW , with bδ∆S= 2π κ bRX + bRW . The departure from the Wald e...

    work page

  3. [3]

    if H H Waξa = 0 and bRX + bRW = 0, thenS 1 = 0, ∆S= 0, andS H =S W

    work page

  4. [4]

    if H H Waξa = 0 but bRX +bRW ̸= 0, then the correc- tion is entirely in the remaining part of− H H ξ·Θ on the restricted slice

    work page

  5. [5]

    if H H Waξa ̸= 0 but bRX +bRW = 0, then the correc- tion is entirely in the non-Wald part of the horizon Noether charge

    work page

  6. [6]

    if H H Waξa ̸= 0 and bRX + bRW ̸= 0, then bothS 1 and ∆Smust be kept inS H. IV. ILLUSTRATIVE MODELS We now apply the preceding decomposition to several static, spherically symmetric black hole solutions. The examples are chosen to separate the different sources in Eq. (52): a regular nonminimally coupled tensor back- ground for which the additional terms ...

    work page

  7. [7]

    J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973)

    work page 1973

  8. [8]

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

    work page 1973

Show all 58 references
  1. [9]

    S. W. Hawking, Particle Creation by Black Holes, Com- mun. Math. Phys.43, 199 (1975), [Erratum: Com- mun.Math.Phys. 46, 206 (1976)]

    1975

  2. [10]

    Jacobson and R

    T. Jacobson and R. C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett.70, 3684 (1993), arXiv:hep-th/9305016

    1993 arXiv

  3. [11]

    Iyer and R

    V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D50, 846 (1994), arXiv:gr-qc/9403028

    1994 arXiv

  4. [12]

    Jacobson, G

    T. Jacobson, G. Kang, and R. C. Myers, Increase of black hole entropy in higher curvature gravity, Phys. Rev. D 52, 3518 (1995), arXiv:gr-qc/9503020

    1995 arXiv

  5. [13]

    Crnkovic and E

    C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, inThree Hundred Years of Gravitation, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge,

  6. [14]

    G. J. Zuckerman, Action principles and global geometry, inMathematical Aspects of String Theory, edited by S.-T. Yau (World Scientific, Singapore, 1987) pp. 259–284

    1987

  7. [15]

    Lee and R

    J. Lee and R. M. Wald, Local symmetries and con- straints, J. Math. Phys.31, 725 (1990)

    1990

  8. [16]

    Compere, An introduction to the mechanics of black holes, in2nd Modave Summer School in Theoretical Physics(2006) arXiv:gr-qc/0611129

    G. Compere, An introduction to the mechanics of black holes, in2nd Modave Summer School in Theoretical Physics(2006) arXiv:gr-qc/0611129

    2006 arXiv

  9. [17]

    R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D48, R3427 (1993), arXiv:gr-qc/9307038

    1993 arXiv

  10. [18]

    Iyer and R

    V. Iyer and R. M. Wald, A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D52, 4430 (1995), arXiv:gr-qc/9503052

    1995 arXiv

  11. [19]

    Jacobson, G

    T. Jacobson, G. Kang, and R. C. Myers, On black hole entropy, Phys. Rev. D49, 6587 (1994), arXiv:gr- qc/9312023

    1994

  12. [20]

    Feng, H.-S

    X.-H. Feng, H.-S. Liu, H. L¨ u, and C. N. Pope, Black Hole Entropy and Viscosity Bound in Horndeski Gravity, JHEP11(11), 176, arXiv:1509.07142 [hep-th]

    arXiv

  13. [21]

    Feng, H.-S

    X.-H. Feng, H.-S. Liu, H. L¨ u, and C. N. Pope, Thermo- dynamics of Charged Black Holes in Einstein-Horndeski- Maxwell Theory, Phys. Rev. D93, 044030 (2016), arXiv:1512.02659 [hep-th]

    2016 arXiv

  14. [22]

    Minamitsuji and K.-i

    M. Minamitsuji and K.-i. Maeda, Black hole thermody- namics in Horndeski theories, Phys. Rev. D108, 084061 (2023), arXiv:2308.01082 [gr-qc]

    2023

  15. [23]

    Fan, Black holes in vector-tensor theories and their thermodynamics, Eur

    Z.-Y. Fan, Black holes in vector-tensor theories and their thermodynamics, Eur. Phys. J. C78, 65 (2018), arXiv:1709.04392 [hep-th]

    2018 arXiv

  16. [24]

    Minamitsuji and K.-i

    M. Minamitsuji and K.-i. Maeda, Black hole thermody- namics in generalized Proca theories, Phys. Rev. D110, 024047 (2024), arXiv:2403.08986 [gr-qc]

    2024

  17. [25]

    An, Notes on thermodynamics of Schwarzschild- like bumblebee black hole, Phys

    Y.-S. An, Notes on thermodynamics of Schwarzschild- like bumblebee black hole, Phys. Dark Univ.45, 101520 (2024), arXiv:2401.15430 [gr-qc]

    2024

  18. [26]

    A. C. Wall and Z. Yan, Linearized second law for higher curvature gravity and nonminimally cou- pled vector fields, Phys. Rev. D110, 084005 (2024), arXiv:2402.05411 [gr-qc]

    2024

  19. [27]

    Comp` ere,Advanced Lectures on General Relativity, Lecture Notes in Physics, Vol

    G. Comp` ere,Advanced Lectures on General Relativity, Lecture Notes in Physics, Vol. 952 (Springer, Cham, Cham, Switzerland, 2019)

    2019

  20. [28]

    S. G. Avery and B. U. W. Schwab, Noether’s second the- orem and Ward identities for gauge symmetries, JHEP 02(02), 031, arXiv:1510.07038 [hep-th]

    arXiv

  21. [29]

    Barnich and F

    G. Barnich and F. Del Monte, Introduction to Classical Gauge Field Theory and to Batalin-Vilkovisky Quanti- zation (2018), arXiv:1810.00442 [hep-th]

    2018 arXiv

  22. [30]

    Barnich, F

    G. Barnich, F. Brandt, and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept.338, 439 (2000), arXiv:hep-th/0002245

    2000 arXiv

  23. [31]

    Racz and R

    I. Racz and R. M. Wald, Extension of space-times with Killing horizon, Class. Quant. Grav.9, 2643 (1992)

    1992

  24. [32]

    P. T. Chrusciel, J. Lopes Costa, and M. Heusler, Station- ary Black Holes: Uniqueness and Beyond, Living Rev. Rel.15, 7 (2012), arXiv:1205.6112 [gr-qc]

    2012 arXiv

  25. [33]

    H. K. Kunduri and J. Lucietti, Classification of near- horizon geometries of extremal black holes, Living Rev. Rel.16, 8 (2013), arXiv:1306.2517 [hep-th]

    2013 arXiv

  26. [34]

    Kalb and P

    M. Kalb and P. Ramond, Classical direct interstring ac- tion, Phys. Rev. D9, 2273 (1974)

    1974

  27. [35]

    Altschul, Q

    B. Altschul, Q. G. Bailey, and V. A. Kostelecky, Lorentz violation with an antisymmetric tensor, Phys. Rev. D81, 065028 (2010), arXiv:0912.4852 [gr-qc]

    2010 arXiv

  28. [36]

    C. A. Hernaski, Spontaneous Breaking of Lorentz Sym- metry with an antisymmetric tensor, Phys. Rev. D94, 105004 (2016), arXiv:1608.00829 [hep-th]

    2016 arXiv

  29. [37]

    Yang, Y.-Z

    K. Yang, Y.-Z. Chen, Z.-Q. Duan, and J.-Y. Zhao, Static and spherically symmetric black holes in gravity with a background Kalb-Ramond field, Phys. Rev. D108, 124004 (2023), arXiv:2308.06613 [gr-qc]

    2023

  30. [38]

    Duan, J.-Y

    Z.-Q. Duan, J.-Y. Zhao, and K. Yang, Electrically charged black holes in gravity with a background Kalb–Ramond field, Eur. Phys. J. C84, 798 (2024), arXiv:2310.13555 [gr-qc]

    2024

  31. [39]

    W. Liu, D. Wu, and J. Wang, Static neutral black holes in Kalb-Ramond gravity, JCAP09(09), 017, arXiv:2406.13461 [hep-th]

  32. [40]

    Liu, S.-P

    J.-Z. Liu, S.-P. Wu, S.-W. Wei, and Y.-X. Liu, Exact black hole solutions in gravity with a background Kalb- Ramond field, JCAP11(11), 056, arXiv:2505.07404 [gr- qc]

  33. [41]

    Xiao and A

    Y. Xiao and A. Zhang, Validity of the background subtraction method for black hole thermodynamics in matter-coupled gravity theories, Phys. Rev. D113, 044047 (2026), arXiv:2511.07209 [gr-qc]

    2026 arXiv

  34. [42]

    V. A. Kostelecky and S. Samuel, Spontaneous Breaking of Lorentz Symmetry in String Theory, Phys. Rev. D39, 683 (1989)

    1989

  35. [43]

    Bluhm and V

    R. Bluhm and V. A. Kostelecky, Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity, Phys. 13 Rev. D71, 065008 (2005), arXiv:hep-th/0412320

    2005 arXiv

  36. [44]

    Q. G. Bailey and V. A. Kostelecky, Signals for Lorentz violation in post-Newtonian gravity, Phys. Rev. D74, 045001 (2006), arXiv:gr-qc/0603030

    2006 arXiv

  37. [45]

    V. A. Kostelecky, Gravity, Lorentz violation, and the standard model, Phys. Rev. D69, 105009 (2004), arXiv:hep-th/0312310

    2004 arXiv

  38. [46]

    Casana, A

    R. Casana, A. Cavalcante, F. P. Poulis, and E. B. Santos, Exact Schwarzschild-like solution in a bumble- bee gravity model, Phys. Rev. D97, 104001 (2018), arXiv:1711.02273 [gr-qc]

    2018 arXiv

  39. [47]

    R. V. Maluf and J. C. S. Neves, Black holes with a cos- mological constant in bumblebee gravity, Phys. Rev. D 103, 044002 (2021), arXiv:2011.12841 [gr-qc]

    2021

  40. [48]

    Liu, W.-D

    J.-Z. Liu, W.-D. Guo, S.-W. Wei, and Y.-X. Liu, Charged spherically symmetric and slowly rotating charged black hole solutions in bumblebee gravity, Eur. Phys. J. C85, 145 (2025), arXiv:2407.08396 [gr-qc]

    2025

  41. [49]

    Liu, S.-P

    J.-Z. Liu, S.-P. Wu, S.-W. Wei, and Y.-X. Liu, Exact Black Hole Solutions in Bumblebee Gravity with Light- like or Spacelike VEVS (2025), arXiv:2510.16731 [gr-qc]

    2025

  42. [50]

    Zhu and H

    J. Zhu and H. Li, Full classification of static spherical vac- uum solutions to bumblebee gravity with general VEVs, Phys. Lett. B876, 140396 (2026), arXiv:2511.03231 [gr- qc]

    2026 arXiv

  43. [51]

    Beltran Jimenez and T

    J. Beltran Jimenez and T. S. Koivisto, Extended Gauss- Bonnet gravities in Weyl geometry, Class. Quant. Grav. 31, 135002 (2014), arXiv:1402.1846 [gr-qc]

    2014 arXiv

  44. [52]

    Bahamonde and M

    S. Bahamonde and M. Ba˜ nados, An exact five dimen- sional Weyl-geometry Gauss-Bonnet black hole, Phys. Lett. B869, 139869 (2025), arXiv:2504.02230 [gr-qc]

    2025

  45. [53]

    Lovelock, The Einstein tensor and its generalizations, J

    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys.12, 498 (1971)

    1971

  46. [54]

    Charmousis, P

    C. Charmousis, P. G. S. Fernandes, and M. Hassaine, Proca theory of four-dimensional regularized Gauss- Bonnet gravity and black holes with primary hair, Phys. Rev. D111, 124008 (2025), arXiv:2504.13084 [gr-qc]

    2025

  47. [55]

    Liu, S.-J

    J.-Z. Liu, S.-J. Yang, C.-C. Zhu, and Y.-X. Liu, D- dimensional black holes in extended Gauss-Bonnet grav- ity (2025), arXiv:2508.04292 [gr-qc]

    2025

  48. [56]

    Alkac, M

    G. Alkac, M. Mesta, and G. Unal, AdS3 black holes with primary Proca hair from a regularized Gauss- Bonnet coupling, Phys. Rev. D112, 084055 (2025), arXiv:2508.03386 [hep-th]

    2025

  49. [57]

    Y. Xiao, Y. Tian, and Y.-X. Liu, Extended black hole thermodynamics from extended Iyer-Wald formalism, Phys. Rev. Lett.132, 021401 (2024), arXiv:2308.12630 [gr-qc]

    2024

  50. [58]

    Xiao, Y.-X

    Y. Xiao, Y.-X. Liu, Y. Tian, and H. Zhang, Explicit and universal formula for thermodynamic volume in extended black hole thermodynamics (2025), arXiv:2512.01916 [gr- qc]

    2025 arXiv

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