Thermodynamic Stability of Schwarzschild-de Sitter Black holes with R\'enyi entropy

Pitayuth Wongjun; Ratchaphat Nakarachinda; Thanawat Anusonthi

REVIEW 6 major objections 5 minor 22 references

Thermodynamic Stability of Schwarzschild-de Sitter Black holes with R\'enyi entropy

T0 review · 6 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a Schwarzschild-de Sitter black hole can be locally and globally thermodynamically stable in an isobaric, fixed-particle-number process once the Rényi non-extensive parameter is promoted to a chemical potential.

desk verdict A coherent Rényi-entropy model that gets SdS stability from a chemical-potential identification, but the identification is assumed, not derived. read the letter →

arxiv 2501.04378 v1 pith:KXZYYXRH submitted 2025-01-08 gr-qc

Thanawat Anusonthi , Pitayuth Wongjun , Ratchaphat Nakarachinda This is my paper · ORCID

classification gr-qc MSC 83C5780A10 PACS 04.70.Dy

keywords Schwarzschild-deSitterblackholeRényientropythermodynamicstabilityheatcapacityGibbsfreeenergychemicalpotentialphasetransitionnon-extensivethermodynamics

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

Schwarzschild-de Sitter black holes, described with Gibbs-Boltzmann entropy, are thermodynamically unstable: the heat capacity is always negative and the Gibbs free energy always positive. The paper argues that switching to Rényi entropy and treating the non-extensive parameter λ as a genuine thermodynamic variable—whose conjugate is an effective particle number Nλ—changes this conclusion. Under an isobaric process at fixed Nλ, a window of horizon radii exists where the heat capacity is positive and the Gibbs free energy negative, making the black hole both locally and globally stable. The same formalism produces first- and second-order phase transitions from a hot gas to a stable black hole, in analogy with the Hawking-Page transition. The upshot is that stable de Sitter black holes, if observed, would favour Rényi statistics over the standard Gibbs-Boltzmann approach.

What carries the argument

The load-bearing structure is the phase-space extension: treating the Rényi non-extensive parameter λ as an independent thermodynamic variable and reading its conjugate Ψλ as a particle number Nλ. Because Ψλ's leading term is proportional to $rH^{3}$, it is extensive, so the first law becomes dm = TR dSbh + VΛ dPΛ + Nλ dμλ. The fixed-Nλ constraint (Eq. 22) then links variations δrH/δλ and is what generates the stable window. The paper uses Euler's theorem on the homogeneous mass function to obtain the Smarr formula, and numerical evaluation of the heat capacity and Gibbs free energy to locate the stable range.

What would settle it

Recompute the heat capacity and Gibbs free energy for the same black hole with λ held constant (the usual Rényi treatment, as in Ref. [3]) instead of imposing the fixed-Nλ constraint of Eq. (22); if the positive-heat-capacity and negative-Gibbs-energy window disappears, the stability is an artifact of the Legendre extension. Alternatively, show that Ψλ is not extensive under scaling of the horizon radius for some λ value, which would break the interpretation of Nλ as a particle number.

Watch

Extended reading notes

Core claim

The central discovery is that the thermodynamic instability of the Schwarzschild-de Sitter black hole is not intrinsic but an artifact of the Gibbs-Boltzmann ensemble. Once the Rényi entropy SR = $λ^{{-1}}$ ln(1 + λ SBH) is used and the phase space is extended so that λ is a chemical potential with conjugate particle number Nλ, the mass becomes a homogeneous function of degree 1/2 in (Sbh, $Λ^{{-1}}$, $λ^{{-1}}$). Imposing a fixed-pressure, fixed-Nλ process, the heat capacity CPΛ,Nλ turns positive and the Gibbs free energy Gbh = m - TR Sbh - μλ Nλ turns negative in a finite horizon-radius interval (numerically for Λ = 0.2, Nλ = 0.3, roughly rH between 1.416 and 1.486). Phase transitions between hot gas and black hole occur at TR = 0.166 (first order) and TR = 0.215 (second order).

Load-bearing premise

The result rests on treating the Rényi non-extensive parameter λ as a real thermodynamic variable whose conjugate is a true particle number; if λ is not an independent degree of freedom, the stable window and phase transitions are artifacts of the extended ensemble.

Editorial extensions

If this is right

In the Rényi ensemble, a Schwarzschild-de Sitter black hole can be stable in an isobaric closed process, which is impossible in the Gibbs-Boltzmann description where CΛ < 0 and GGB > 0 always.
For Λ = 0.2 and Nλ = 0.3, stable black holes have horizon radii between about 1.416 and 1.486, with a mass roughly 140 times the Earth's mass.
The Gibbs free energy versus temperature diagram shows a first-order phase transition from hot gas to black hole at TR ≈ 0.166 and a second-order transition at TR ≈ 0.215.
The entropy that yields this stability is the Rényi entropy, whose logarithmic map ensures consistency with the zeroth law of thermodynamics, unlike the raw Tsallis entropy.

Reading between the lines

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

If stable de Sitter black holes are ever observed, their horizon temperatures and masses could be used to estimate the Rényi parameter λ, connecting black-hole thermodynamics to non-extensive statistical mechanics.
The authors leave open the heat capacities at fixed volume and fixed chemical potential (CP,μ, CV,N, and CV,μ); computing these would show whether the stable window survives in open ensembles or is specific to the fixed-Nλ isobar.
Because the argument hinges on the extensivity of Nλ, a microscopic derivation of λ as a chemical potential from quantum gravity microstates would either validate or falsify the phase-space extension.
A direct extension is to include charge or rotation, checking whether the stable window persists when the Smarr formula gains additional work terms.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new thing here is the explicit reading of the Rényi parameter λ as a chemical potential and the isobaric fixed-N stability analysis that follows. That combination produces a stable Schwarzschild-de Sitter branch and Hawking-Page-like phase transitions, which is not in the earlier Rényi-stability papers. The Smarr and first-law derivations are explicit and self-consistent under Euler's theorem, and the recovery of the known Gibbs-Boltzmann instability in the λ→0 limit is a good consistency check.

The soft spot is exactly where the stress-test note lands. Nλ is identified as a particle number because the leading term in its expansion scales like r_H^3. Volume scaling is not enough to make something a conserved charge. The paper offers no physical mechanism or independent symmetry that would couple a chemical potential to the Rényi parameter in Einstein gravity. The fixed-N constraint δNλ=0 is then an input that selects a curve in extended phase space, and the stable interval and phase transitions are consequences of that constraint. The paper is transparent about this being an interpretation, but it does not justify the interpretation beyond extensivity. That is a real gap, and it makes the central claim conditional.

The numerics are thin: one parameter choice (Λ=0.2, Nλ=0.3), no code, no error bars, no robustness scan. A referee should ask for a scan over Λ and Nλ and for a check that the stable range is not a numerical artifact. There are also minor typos (e.g., "0th order" in the text likely means "2nd order") and some awkward phrasing, but nothing that affects the logic.

Still, the paper is a serious model calculation. The question is well-posed within the Rényi program, the derivations are explicit, and the limitations are honestly acknowledged in the conclusion. It deserves peer review, not desk rejection. A competent referee can push on the Nλ identification and request better numerics, but the paper should get the chance to respond.

I would cite it only if I were working actively in Rényi black hole thermodynamics, which I am not. But I would bring it to a reading group discussion about extended phase spaces and the meaning of chemical potentials in black hole thermodynamics.

Referee Report

6 major / 5 minor

Summary. The paper studies the thermodynamic stability of Schwarzschild-de Sitter black holes using Rényi entropy. The authors extend the thermodynamic phase space by treating the non-extensive parameter λ as a thermodynamic variable whose conjugate, Ψλ = ∂m/∂λ, is interpreted as a particle number Nλ and λ as a chemical potential. After deriving the Smarr formula and first law in this extended ensemble, they impose a process at fixed pressure Λ and fixed particle number Nλ, and report that for a specific choice (Λ=0.2, Nλ=0.3) there exists a range of horizon radii where the heat capacity is positive and the Gibbs free energy is negative, indicating local and global stability. They also identify phase transitions between a hot gas and the black hole. The paper concludes that Rényi statistics can stabilize the Schwarzschild-de Sitter black hole, in contrast to the Gibbs-Boltzmann description.

Significance. If the central claim were established, it would be a notable contribution to black hole thermodynamics: it would offer a concrete scenario in which non-extensive Rényi entropy stabilizes a black hole in de Sitter spacetime and would extend the black-hole-chemistry framework to a new ensemble. The paper contains an explicit, self-consistent derivation of the Smarr formula and first law under its stated assumptions, which is a useful technical piece. However, the significance is severely limited by three facts: the global-stability analysis uses a thermodynamic potential that does not correspond to the claimed fixed-N process; the identification of Nλ as a physical particle number is supported only by an extensivity heuristic and not by a conserved charge; and the central numerical result rests on a single parameter point with no reproducibility data. These issues leave the main claim unsupported in its current form.

major comments (6)

[§4, Eq. (23)] The global stability criterion is applied to G_bh = m - T_R S_bh - μλ Nλ, but the process described in the text fixes pressure and number of particles. For such a process, the appropriate Gibbs free energy in the enthalpy representation is G = m - T_R S_bh, because the first law (18) identifies m as the enthalpy (d m = T dS + V dP + μ dN). Subtracting μλ Nλ changes the ensemble to fixed chemical potential, and the condition G < 0 does not establish global stability for fixed Nλ. Please either use G = m - T_R S_bh for the fixed-N analysis or explicitly state that the stability and phase transitions refer to the grand canonical ensemble; the latter would contradict the fixed-N constraint in Eq. (22).
[§4, Eq. (22) and identification of Nλ] The particle number Nλ is defined as ∂m/∂λ and is a derived function of r_H and Λ, not an independent conserved charge. The only evidence for interpreting Ψλ as a particle number is the leading-order r_H^3 scaling in Eq. (16), but volume scaling alone does not make a quantity a particle number; a chemical potential must couple to a conserved charge, and no such charge is identified for the Rényi parameter in Einstein gravity. Imposing δNλ = 0 therefore selects a curve (r_H, λ) in the extended phase space rather than describing a physical constraint. The authors should either identify a conserved charge associated with λ or test whether the claimed stability survives when λ is held fixed instead of Nλ.
[§4, Figures 1 and 2] The central numerical claim is supported by a single parameter choice, Λ = 0.2 and Nλ = 0.3, with no code, data, error estimates, or scan over the parameter space. Because the fixed-Nλ constraint is what produces the stable interval, it is essential to show that the stability is robust to order-of-magnitude changes in these parameters and to compare with the case λ = constant (which would correspond to the question of whether Rényi entropy itself, rather than the extra constraint, stabilizes the black hole). Without such a robustness analysis, the claim that Rényi entropy stabilizes the Schwarzschild-de Sitter black hole is not established.
[§4, Eqs. (14)-(15)] There is a sign inconsistency in the Legendre transformation. If E = m - P_Λ V_Λ - Ψλ λ and the first law is d m = T_R dS_bh + V_Λ dP_Λ + Ψλ dλ, then dE = T_R dS_bh - P_Λ dV_Λ - Ψλ dλ, not dE = T_R dS_bh - P_Λ dV_Λ - λ dΨλ as written. This inconsistency affects the subsequent identification of μλ = -∂E/∂Nλ = λ in Eq. (17) and the form of the first law in Eq. (18). Please correct the sign and re-derive the stability conditions accordingly.
[§4, numerical method] The paper states that the heat capacity and Gibbs free energy are computed numerically using condition (22), but it does not provide the explicit expressions used, the numerical algorithm, the explored ranges of λ and r_H, or the numerical values underlying Figures 1 and 2. This prevents reproduction and independent verification. Please include the relevant formulas or provide a code repository.
[§4 and §5, phase transition orders] The text in Section 4 refers to a '2nd phase transition' at the cusp, while the Conclusion states that the higher-temperature transition is a '0th order' transition; Fig. 2 also labels a point TR = 0.211 that is not fully explained. Please clarify the order and location of each phase transition and make the terminology consistent.

minor comments (5)

[Abstract] The phrase 'a black holes is gravitational object' should be 'a black hole is a gravitational object'.
[§1] The sentence 'One finds that the Schwarzschild-de Sitter black hole is thermodynamically unstable by using Gibbs-Boltzmann statistics' would read more clearly as 'It has been found...'
[§3] The name 'Schwarzschild-Tangherlini' is used for the Schwarzschild-de Sitter metric; the Tangherlini solution refers to higher dimensions, so this naming is inaccurate.
[References] Reference [6] cites only the title and journal of Smarr's paper but not the author; it should read 'L. Smarr, Mass Formula for Kerr Black Holes, Phys. Rev. Lett. 30, 71 (1973)'.
[§4, Fig. 2] The figure caption and the text should be cross-checked: the text mentions TR = 0.166, 0.211, and 0.215, but the explanation of the cusp and the stable range would benefit from a direct reading of the corresponding horizon radii.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stable branch is computed from the Rényi-smarr framework, not imposed as an input.

full rationale

The paper's derivation is self-contained in the circularity sense. It begins with the Rényi entropy definition S_bh = (1/lambda) ln(1 + lambda S_BH), constructs the mass function (7), obtains the Smarr relation via Euler's theorem (9)-(11), and introduces the conjugate pair (lambda, N_lambda = partial m/partial lambda) through the Legendre transformation (14)-(18). The stability calculation then fixes Lambda and N_lambda, uses delta N_lambda = 0 in Eq. (22) to determine the curve delta r_H/delta lambda, and evaluates C_{P_lambda,N_lambda} and G_bh as functions of r_H. Positivity of the heat capacity and negativity of the Gibbs free energy are outputs of the numerical evaluation, not assumptions inserted into the model. The values Lambda=0.2 and N_lambda=0.3 are presented as an existence example, not fitted to any data, and the stable horizon-radius interval is not encoded in the choice of those values. The self-citations [8,9,12] supply the phase-space-extension formalism and earlier Rényi-stability results, but they are parameter-free developments whose stated assumptions do not include the specific stable-branch claim of this paper; under the reviewing rules they count as independent support rather than circularity. The closing caveat that other heat capacities and compressibility remain to be investigated is an acknowledged scope limitation, not a circular step. Concerns that N_lambda is not a conserved charge or that the hot-gas reference has N_lambda = 0 and therefore cannot coexist with a fixed-N_lambda black hole are physical correctness objections to the extended ensemble, not cases where a prediction reduces to its input by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central claim depends on three hand-set values (Lambda, Nlambda) and on treating the Rényi parameter lambda as a thermodynamic variable. The latter is an ad hoc extension of the phase space, and the interpretation of the conjugate as particle number is heuristic. No new particles or fields are introduced, only a new thermodynamic role for lambda.

free parameters (3)

Cosmological constant Lambda = 0.2 (chosen for numerical example)
Set by hand in Section 4 to produce Figure 1; the stable range and transition temperatures depend on this choice.
Particle number Nlambda = 0.3 (chosen for numerical example)
Set by hand in Section 4; the conjugate of lambda is interpreted as particle number but no physical origin or integer constraint is given.
Renyi non-extensive parameter lambda = not fixed; varies with horizon radius under fixed-N condition (Eq. 22)
The central framework treats lambda as a thermodynamic variable or chemical potential; its value is not determined by independent physics.

assumptions (5)

standard math Euler's homogeneous function theorem applies to the mass function m(Sbh, Lambda^-1, lambda^-1)
Used in Eqs. (9)-(11) to derive the Smarr formula.
domain assumption Bekenstein-Hawking entropy can be identified with Tsallis entropy, and Rényi entropy via logarithmic map is the appropriate equilibrium entropy for black holes
Stated in Section 2, Eqs. (1)-(4), following Refs. [3,5,9,12]; underpins all subsequent thermodynamics.
ad hoc to paper The non-extensive parameter lambda must be promoted to a thermodynamic variable with conjugate Nlambda
Introduced in Section 3 via Legendre transform (Eq. 14); no derivation from statistical mechanics or gravity, and the stability result depends on it.
ad hoc to paper Nlambda=partial m/partial lambda can be interpreted as number of particles because its leading term scales as r^3
Based on Taylor expansion Eq. (16); the analogy to particle number is heuristic and affects the fixed-N constraint (Eq. 22).
domain assumption Hot gas state has vanishing thermodynamic variables and zero Gibbs free energy
Assumed in Section 4 before Eq. (20) to define global stability relative to hot gas.

invented entities (2)

Chemical potential mu_lambda = lambda for the Rényi non-extensive parameter
purpose: Treats the fixed entropy index as a thermodynamic variable to extend phase space and obtain a first law (Eqs. 17-18)
No independent falsifiable handle; the identification is made by Legendre transform and Taylor expansion in Section 3. If wrong, the stable range in Figure 1 loses its meaning.
Particle number Nlambda = partial m/partial lambda
purpose: Conjugate variable to lambda; fixed during the isobaric process, used to define closed system stability in Eq. (22)
Interpreted as number of particles due to r^3 scaling, but no microscopic particle counting is given; N=0.3 is non-integer.

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Pith. "Pith review of Thermodynamic Stability of Schwarzschild-de Sitter Black holes with R\'enyi entropy." pith.science (2026). https://pith.science/paper/KXZYYXRH

@misc{pith2026250104378,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic Stability of Schwarzschild-de Sitter Black holes with R\'enyi entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXZYYXRH}},
  note         = {Machine review of arXiv:2501.04378}
}

read the original abstract

Even though, classically, a black holes is gravitational object, it can be treated as a thermodynamic object when quantum effect is taken into account. It is found that by using Gibbs-Boltzmann entropy, thermodynamic system associated Schwarzschild-de Sitter black hole is unstable while it is stable under description of R\'enyi entropy. According to R\'enyi entropy, the thermodynamic phase space needed to be extended. Specifically, the non-extensive parameter must be treated as a thermodynamic variable. In this work, we investigate the thermodynamic stability of Schwarzschild-de Sitter black hole with R\'enyi entropy by treating non-extensive parameter as chemical potential. We found that it is possible to obtain the stability of the black hole under the process with fixing pressure, temperature and number of particles. We also found that there exist the phase transitions from the hot gas to the stable black hole in such the process. Therefore, if such a black hole is possibly observed, the non-extensive effect of R\'enyi statistics may provide a physical insight beyond the standard approach to black hole thermodynamics.

Figures

Figures reproduced from arXiv: 2501.04378 by the authors.

**Figure 1.** Gibbs free energy and heat capacity horizon radius in which the black hole is thermodynamically stable is represented by the pink highlight. By comparing with the horizon radius, we can track back to the mass of the black hole. For example, a given cosmological constant is Λ = 0.2. The Schwarzschild-de Sitter black hole is thermodynamically stable with a mass around 140 times the mass of the earth. In [PITH_FULL_IM… view at source ↗

**Figure 2.** Gibbs free energy and Temperature TR = 0.215, a cusp of the graph, the heat capacity is changed its sign corresponding to 2nd order phase transition. After that the system will evolve to a certain temperature suppose to be in the range of r = 1.416 to r = 1.486 denoted by the pink line as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

R´ enyi entropy

Introduction & Motivation After Hawking temperature and Bekenstein-Hawking entropy were explored [1, 2], a black hole can be treated as a thermodynamic object. Thus, the stability of a black hole in thermody- namic aspect is worthwhile to investigate. By considering heat capacity and Gibbs free energy, one found that the Schwarzschild black hole is thermo...

arXiv 2025

[2] [2]

non-extensive parameter

Black holes with R´ enyi entropy By comparing black body’s radiation based on Gibbs-Boltzmann statistics, Hawking obtained the temperature of a black hole called Hawking temperature [1]. Thus, the corresponding entropy which known as the Bekenstein-Hawking (BH) entropy [2] can be expressed as SBH = A 4 , (1) where A is the surface area at the black hole’s...

work page

[3] [3]

R´ enyi temperature

Thermodynamic system of Schwarzschild-de Sitter black hole The Schwarzschild-de Sitter black hole solution can be written as ds2 = −f (r)dt2 + f (r)dr2 + r2dΩ2, (5) with f (r) = 1 − 2m r − Λ 3 r2 where Λ > 0. (6) The parameter m is the ADM mass [10] and Λ is the cosmological constant. At the event horizon, f (rH ) = 0, the mass of the black hole can be wr...

work page

[4] [4]

• Local stability Local stability is related to the responsiveness of a system when we perturb it

Thermodynamic stability In equilibrium thermodynamics, the stability analysis can be divided into two branches, namely local and global stability [16, 17]. • Local stability Local stability is related to the responsiveness of a system when we perturb it. The local stability condition is characterized by positive heat capacity, CX = δQ δT X > 0, (19) where...

work page

[5] [5]

response function

Conclusion & Discussion To study the non-extensive behavior of entropy of a black hole, we have treated Bekenstein- Hawking entropy as the non-extensive Tsallis entropy. In order to describe a black hole by equilibrium thermodynamics, the entropy of a system should be consistent with the 0th law of thermodynamics. The entropy should be additive. R´ enyi e...

work page

[6] [6]

Acknowledgement This research has received funding support from the NSF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation [ grant number B37G660013 ]

work page

[7] [7]

Particle creation by black holes

Hawking SW. Particle creation by black holes. Commun Math Phys. 1975;43:199-220. [Erra- tum:Commun.Math.Phys. 46, 206 (1976)]

work page 1976

[8] [8]

Black holes and entropy

Bekenstein JD. Black holes and entropy. Phys Rev D. 1973;7:2333-46

work page 1973