Thermodynamics of deformed AdS-Schwarzschild black holes in the presence of Thermal fluctuations

Dhruba Jyoti Gogoi; Jyatsnasree Bora; Poppy Hazarika; Ranjan Changmai

REVIEW 4 major objections 4 minor 69 references

Thermodynamics of deformed AdS-Schwarzschild black holes in the presence of Thermal fluctuations

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that increasing the deformation parameter alpha of an AdS-Schwarzschild black hole raises the Hawking-Page critical temperature, and that thermal-fluctuation corrections mainly change small black holes without altering…

desk verdict The thermal-correction results are built on a factor-of-two algebraic slip in Eq. (23), so the paper needs a full algebra recheck before its numbers can be trusted. read the letter →

arxiv 2501.15629 v2 pith:53VX3JNP submitted 2025-01-26 hep-th gr-qc

Dhruba Jyoti Gogoi , Poppy Hazarika , Jyatsnasree Bora , Ranjan Changmai This is my paper · ORCID

classification hep-thgr-qc MSC 83C5780A10 PACS 04.70.-s04.70.Dy

keywords deformedAdS-SchwarzschildblackholethermalfluctuationscorrectedentropyHawking-PagephasetransitionheatcapacitystabilityGibbsfreeenergy

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

This paper argues that a deformed AdS-Schwarzschild black hole — a spherically symmetric black hole in anti-de Sitter spacetime whose metric carries an extra deformation term $\alpha$ — becomes thermally more stable as the deformation parameter grows. Concretely, it claims that larger $\alpha$ raises the Hawking-Page critical temperature, the point where the Gibbs free energy changes sign and the black hole switches between a stable and an unstable phase. The paper then adds thermal-fluctuation corrections to the entropy, controlled by two parameters $\beta_1$ and $\beta_2$, and claims these corrections significantly change the thermodynamics of small black holes while leaving large black holes nearly untouched. Corrected heat capacity maps stable and unstable horizon-radius regions; sign changes mark the boundaries. If these claims are correct, both the deformation and the thermal corrections act as stabilizing influences, broadening the temperature range in which the black hole is thermodynamically preferred.

What carries the argument

The carrying mechanism is the corrected entropy formula $S = S_0 - \frac{\beta_1}{2}\log(S_0 T_H^2) + \frac{\beta_2}{S_0}$, with uncorrected area entropy $S_0 = \pi r_+^2$, fed into the Hawking temperature $T_H$ computed from the deformed metric function $F(r)=1-\frac{2M}{r}+\frac{r^2}{l^2}+\frac{\alpha(\beta^2+3r^2+3\beta r)}{3r(\beta+r)^3}$. This formula turns every thermodynamic potential into a function of $\alpha,\beta,\beta_1,\beta_2$; the paper then uses the sign of the corrected Gibbs free energy against $T_H$ to locate the Hawking-Page transition and the sign of the corrected specific heat to locate stable and unstable regions.

What would settle it

Compute the canonical partition function directly from the deformed metric and compare the small-horizon density of states with the expansion behind Eq. (22); if the coefficients or functional form differ, the claimed correction-driven stabilization of small black holes fails. A simpler check is numerical: scan the horizon-radius domain with fixed $\alpha,\beta,\beta_1,\beta_2$ and look for a radius where the corrected heat capacity is positive while the uncorrected one is negative; if no such radius exists, the thermal corrections are not doing the stabilizing work attributed to them.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is an extension: starting from a deformed AdS-Schwarzschild metric with deformation parameter $\alpha$, it applies a standard second-order thermal-fluctuation corrected entropy $S=S_0-\frac{\beta_1}{2}\log(S_0 T_H^2)+\frac{\beta_2}{S_0}$ and derives closed-form corrections to enthalpy, volume, Helmholtz and Gibbs free energy, internal energy, and heat capacity. The pattern that emerges is that $\alpha$ pushes the Hawking-Page transition to higher temperatures; $\beta$ moves it slightly downward; and the correction parameters $\beta_1,\beta_2$ alter small-horizon behaviour, make the corrected Gibbs free energy more negative, and shift physical limitation points without changing the location of second-order phase transitions. The paper reads the sign of the corrected heat capacity as the stability criterion: positive regions are stable, negative regions unstable, with sign changes at physical limitation or divergence points.

Load-bearing premise

The load-bearing premise is that the entropy correction is a logarithmic term plus an inverse-entropy term with free coefficients $\beta_1,\beta_2$; the paper does not derive this formula for the deformed metric or check that the expansion converges.

Editorial extensions

If this is right

Higher values of $\alpha$ raise the Hawking-Page critical temperature, so the deformed black hole remains thermally stable at higher temperatures.
Higher values of $\beta$ slightly lower the critical temperature, so smaller $\beta$ acts in the same stabilizing direction as larger $\alpha$.
Thermal-fluctuation corrections $\beta_1,\beta_2$ modify the thermodynamics of small black holes, including a more negative corrected Gibbs free energy, while large black holes are essentially unaffected.
The corrected heat capacity has sign changes at physical limitation and divergence points; positive regions correspond to thermal stability and negative regions to instability.
Second-order phase transitions are preserved under the thermal corrections, which only shift limiting points slightly.

Reading between the lines

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

A direct extension the paper leaves implicit is a microphysical derivation of $\beta_1$ and $\beta_2$ for this deformed metric instead of treating them as free parameters.
If the corrections matter mainly for small horizons, their effects would be most visible in the final stages of black-hole evaporation, possibly as a shift in the evaporation timescale or a remnant-like behaviour.
Because the Hawking-Page transition temperature shifts with $\alpha$ while the second-order transition does not, the model offers a way to separate deformation effects from fluctuation effects in the phase diagram of AdS black holes.
One could test the stability claim directly by checking whether the corrected heat capacity becomes positive at radii where the uncorrected heat capacity is negative for the same parameter values; the paper does not report such a side-by-side comparison.

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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

Quick take: the paper's central thermal-correction results are built on an algebraic slip. Eq. (23) claims to be the explicit form of Eq. (22) for this metric, but it isn't. Substituting S0 = πr_+^2 and TH from Eq. (9) into Eq. (22) gives log-coefficients (β1/2)log(16π) + 4β1 log(β+r_+) − (β1/2)log(N^2). Eq. (23) instead has β1 log(16π) + 8β1 log(β+r_+) − β1 log(N^2), i.e. every β1 term is twice as large. All corrected quantities—Hc, Vc, Fc, Uc, Gc, Cc—are downstream of Eq. (23), so the quantitative claims about how β1, β2 affect small black holes are computed from the wrong expression.

That said, the paper does something legitimate: it takes a known deformed AdS-Schwarzschild metric (Ref. [38]) and a known entropy-correction formula (Refs. [40,41]) and works out the corrected potentials explicitly. The tables of physical limitation points and divergences are useful, and the qualitative statement that increasing α raises the Hawking-Page temperature is plausible, since α enters TH and the mass directly; that part likely survives the correction.

The soft spots are real and load-bearing. Besides the factor-of-two error, the uncorrected limits don't close: setting β1=β2=0 in Hc (Eq. 25) does not return H (Eq. 13), and U in Eq. (15) does not follow from H−PV using the given H and V. That's a consistency check the authors state they performed—they explicitly say β1=β2=0 recovers S0—but it fails for Hc. The correction parameters β1, β2 are also introduced as free parameters with no derivation for this spacetime and no check that the canonical-ensemble expansion converges. So the central quantitative claims are unsupported as written.

The paper is readable and the literature usage is fine; the self-citations are not load-bearing. It's a modest extension of an established program, not a new mechanism.

Who is this for? People working on thermal-fluctuation corrections to black hole thermodynamics in deformed gravity settings. They'd get a worked example, but they should not quote the β1,β2-dependent numbers until the algebra is redone.

Recommendation: this deserves a serious referee, because the error is concrete and fixable and the paper sits in an active literature. But the referee should be asked to verify every equation from Eq. (23) onward. If the authors redo the algebra and the limits close, the paper can be a useful parameter study.

Referee Report

4 major / 4 minor

Summary. This paper studies the thermodynamics of the deformed AdS-Schwarzschild black hole of Ref. [38]. Starting from the metric (6), the authors compute the Hawking temperature (9), the Bekenstein-Hawking entropy S0, and the enthalpy and internal energy (13)-(15). They then import the thermal-fluctuation entropy correction (22), write an explicit corrected entropy Sc in Eq. (23), and use it to derive corrected enthalpy Hc, volume Vc, Helmholtz free energy Fc, internal energy Uc, Gibbs free energy Gc, and specific heat Cc in Eqs. (25)-(35). Numerical plots and tables are used to argue that larger deformation alpha raises the Hawking-Page temperature, that the control parameter beta acts oppositely, that the corrections beta1 and beta2 significantly affect small black holes but not large ones, and that second-order phase transitions are unchanged.

Significance. If correct, the paper would be an incremental but useful addition to the literature applying the standard entropy-correction program to a deformed AdS-Schwarzschild background. The explicit closed-form expressions allow direct verification, and the beta1=beta2=0 limits are checked in several places, which is a strength. However, the load-bearing input is the imported correction formula Eq. (22), and the paper neither derives it for this spacetime nor checks the convergence of the expansion. The algebraic mismatches described below mean that the quantitative thermal-correction results are not currently consequences of the stated model. The broader qualitative claim that alpha shifts the Hawking-Page temperature may survive the required recomputation, since alpha enters through the uncorrected temperature and mass, but the paper's numerical values and beta1-dependent statements need re-examination. The paper does not include machine-checked proofs or reproducible code; verifiability rests on hand-written expressions.

major comments (4)

[Section IV, Eqs. (22)-(23)] Eq. (23) is not the specialization of Eq. (22) to the deformed metric. Using S0 = pi r_+^2 and TH from Eq. (9), one finds S0 TH^2 = N^2 / [16 pi (beta + r_+)^8], where N is the numerator in Eq. (9). Substituting this into Eq. (22) gives Sc = pi r_+^2 + (beta1/2) log(16 pi) + 4 beta1 log(beta + r_+) - (beta1/2) log(N^2) + beta2/(pi r_+^2). Eq. (23), however, has beta1 log(16 pi) + 8 beta1 log(beta + r_+) - beta1 log(N^2), i.e., every beta1-dependent logarithmic coefficient is twice the value prescribed by Eq. (22). Since Hc, Vc, Fc, Uc, Gc, and Cc in Eqs. (25)-(35) are all built from Eq. (23), all quantitative statements about the beta1 dependence of thermal corrections are computed from a different correction term than the one the paper claims to use; they must be recomputed using the Sc that actually follows from Eq. (22).
[Section III and Section V, Eqs. (13), (15), (25)] The corrected enthalpy Eq. (25) does not reduce to the uncorrected enthalpy Eq. (13) in the limit beta1 = beta2 = 0. Setting beta1 = beta2 = 0 in Eq. (25) gives Hc = (1/6)[alpha(beta^2 + 3 beta r_+ + 3 r_+^2)/(beta + r_+)^3 - Lambda r_+^3 + 3 r_+], whereas Eq. (13) contains the additional terms 3 beta and -Lambda(beta^3 + r_+^3). Relatedly, Eq. (15) is not the U = H - P V consequence of Eqs. (13), (14), and (8): that combination contains a term -Lambda beta^3/6 that is absent from Eq. (15). Thus the uncorrected thermodynamic identities are internally inconsistent, and the corrected potentials are not a controlled beta1,beta2 -> 0 limit of the uncorrected sector.
[Section IV, Eq. (22)] The load-bearing premise is the thermal-fluctuation formula Eq. (22), quoted from Refs. [40,41] without derivation for the deformed AdS-Schwarzschild solution. The expansion in Eqs. (19)-(20) is a saddle-point/Laplace-inversion expansion around the equilibrium inverse temperature; its validity requires the higher cumulants of the entropy to be small in the relevant parameter range. This is not checked for the present metric and for the values of alpha, beta, beta1, and beta2 used in the plots. Because beta1 and beta2 are free parameters and all subsequent thermodynamic potentials inherit Eq. (22), the quantitative claims about how thermal corrections affect small black holes are conditional on this imported formula. Please either derive Eq. (22) for this spacetime from the partition function in Eqs. (16)-(20), or state explicitly the range of r_+, beta, alpha for which the expansion is controlled and verify that the chosen parameters lie in that range.
[Section VI, Figs. 11-12] The central claim that higher alpha raises the Hawking-Page critical temperature is presented through plots of G and Gc versus TH, with r_+ as an implicit parameter. The paper does not provide numerical values of the zero-crossing temperatures for the parameter choices in Fig. 11, nor does it state whether the thermal-AdS background free energy is included in the comparison. In the standard Hawking-Page analysis the transition is determined by the free-energy difference between the black hole and thermal AdS, not merely by the sign of G for the black hole alone. Please give the explicit critical temperatures or the free-energy comparison used to read off the transition, so that the shift claimed in the abstract and conclusions is quantitatively supported.

minor comments (4)

[Fig. 3 caption] The caption reads 'Lambda=0-.002', which should be 'Lambda = -0.002'.
[Tables I and II and Section VI] The phrase 'physical limitation points' is used repeatedly but is never defined; the authors should define it explicitly, e.g., as the zeros of the denominator of the specific heat or the points where the heat capacity changes sign.
[Bibliography] References [41] and [56] are the same paper (Pourhassan and Faizal, Nucl. Phys. B 913, 834 (2016)); this duplicate entry should be removed.
[Notation, Eqs. (25)-(35)] The notation is confusing because the correction parameter beta2 and the square beta^2 are typeset almost identically; using an explicit subscript, for example beta_2, in all displayed equations would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all corrected thermodynamic quantities are derived from the explicitly imported entropy-correction formula, with no fitted parameter called a prediction and no load-bearing self-citation.

full rationale

The paper's derivation chain is a straightforward application of an imported entropy-correction ansatz. Equation (22), S = S0 - (beta1/2) log(S0 T_H^2) + beta2/S0, is taken from Refs. [40,41], and every downstream quantity (Sc, Hc, Vc, Fc, Uc, Gc, Cc) is obtained from it by quadrature and standard thermodynamic identities. No parameter is fitted to a data subset and then re-predicted, and no normalization or self-consistency condition forces the reported conclusions. The claim that thermal corrections matter mainly for small black holes is a direct property of the assumed 1/S0 and logarithmic correction terms, but the paper presents these as consequences of the adopted ansatz rather than as independent empirical predictions, so this is derivation from an assumption, not circularity. The self-citations (Refs. [7,19,43]) appear only as background motivation and are not load-bearing; the metric comes from Ref. [38] and the correction formula from Refs. [40,41], none of which are the present authors' own uniqueness claims. A separate mathematical issue exists: Eq. (23), as printed, has beta1-dependent logarithmic coefficients that are twice the values obtained by substituting S0 and T_H into Eq. (22), so the quantitative thermal-fluctuation results are computed from a misimplemented Sc. This is a correctness/typographical concern, not a circularity, and it does not change the circularity verdict.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The calculation relies on four chosen model parameters (alpha, beta, beta1, beta2) and on imported formulas for the deformed metric and the entropy correction. None of the parameters is measured or constrained, and no new entity is introduced.

free parameters (4)

alpha (deformation parameter)
Controls the deformation terms in the metric; chosen in plots over ranges such as 0 to 2 with no independent determination or constraint.
beta (control parameter)
Controls the energy-density profile in Eq. (5); chosen in plots over ranges such as 0 to 0.9 with no physical constraint.
beta1 (first-order thermal correction coefficient)
Coefficient of the logarithmic entropy correction in Eq. (22); treated as a free parameter and assigned values such as 0.1 to 0.7.
beta2 (second-order thermal correction coefficient)
Coefficient of the inverse-entropy correction in Eq. (22); treated as a free parameter and assigned values such as 0.1 to 0.6.

assumptions (4)

domain assumption The deformed AdS-Schwarzschild metric in Eq. (6), including the alpha-dependent terms, is a valid solution of the modified Einstein equations.
The metric is taken from Ref. [38]; the paper does not re-derive or verify the field equations or energy conditions.
domain assumption The canonical-ensemble entropy correction S = S0 - (beta1/2) log(S0 TH^2) + beta2/S0 applies to this black hole.
Imported from Refs. [40,41]; no derivation or domain-of-validity check for deformed AdS black holes is provided.
domain assumption Higher-order terms in the Taylor expansion of the entropy are negligible.
Eq. (19) is truncated at second order with no estimate of the remainder.
domain assumption Standard thermodynamic identities, V = (dH/dP)_S and C = T dS/dT, remain valid for the corrected quantities.
Used in Eqs. (26) and (34); standard in this literature but not justified when entropy is corrected.

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Cite this review

Pith. "Pith review of Thermodynamics of deformed AdS-Schwarzschild black holes in the presence of Thermal fluctuations." pith.science (2026). https://pith.science/paper/53VX3JNP

@misc{pith2026250115629,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of deformed AdS-Schwarzschild black holes in the presence of Thermal fluctuations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53VX3JNP}},
  note         = {Machine review of arXiv:2501.15629}
}

abstract

This paper examines the thermodynamic properties and stability of deformed AdS-Schwarzschild black holes, focusing on the effects of deformation ($\alpha$) and thermal correction parameters ($\beta_1$, $\beta_2$) on phase transitions and heat capacity. The results show that higher $\alpha$ values raise the Hawking-Page critical temperature, enhancing thermal stability. Thermal corrections significantly affect smaller black holes but minimally impact larger ones, leaving second-order phase transitions unchanged. Heat capacity analysis identifies stability regions, with sign changes marking instability. These findings highlight the role of deformation and thermal corrections in black hole stability, offering insights for extending our understanding of black hole thermodynamics.

Figures

Figures reproduced from arXiv: 2501.15629 by the authors.

**Figure 2.** FIG. 2: Variation of the Hawking temperature with the black hole horizon radius [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Corrected entropy vs. the black hole horizon radius [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: FIG. 4: Corrected entropy vs. the black hole horizon radius [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5: Variation of the corrected Helmholtz energy of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6: Variation of the corrected Helmholtz energy of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7: Variation of the corrected internal energy of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8: Variation of the corrected internal energy of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: FIG. 9: Variation of the corrected Gibbs free energy of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10: Variation of the corrected Gibbs free energy of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11: Variation of the Gibbs free energy with the Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 12.** Figure 12: FIG. 12: Variation of the corrected Gibbs free energy with the Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

**Figure 13.** Figure 13: FIG. 13: Variation of the Gibbs free energy with the Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 14.** Figure 14: FIG. 14: Variation of the corrected specific heat of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

**Figure 15.** Figure 15: FIG. 15: Variation of the corrected specific heat of the black hole with the event horizon radius [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]

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.

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