We compare two thermodynamic prescriptions associated with the same static, spherically symmetric four-dimensional Einstein--Gauss--Bonnet anti-de Sitter black hole. Model A uses the usual Hawking temperature obtained from the surface gravity and consequently yields an entropy with a logarithmic correction. Model B preserves the shift symmetry of the scalar field at the level of the thermodynamic charges, recovers the Bekenstein--Hawking area law, and requires a modified temperature for consistency with the first law. We derive the equations of state, Gibbs free energies, heat capacities, and critical quantities of both prescriptions in the extended phase space. The two models display the same qualitative Van der Waals phase structure, but their critical scales and local stability regions differ quantitatively. In the uncharged case, Model A has a larger critical horizon radius and lower critical temperature and pressure than Model B. A representative charged example further shows that the fixed ratios between the critical quantities of the two prescriptions in the neutral sector do not persist when Q≠0.
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Thermodynamics of Four-Dimensional Einstein--Gauss--Bonnet Black Holes with a Modified Temperature
Author Information
Received: 12 Aug 2026 | Revised: 09 Sep 2026 | Accepted: 11 Sep 2026 | Published: 18 Sep 2026
Abstract
Keywords
black hole thermodynamics | Einstein--Gauss--Bonnet gravity | extended phase space | classical theories of gravity | phase transitions
References
- 1.
Bekenstein, J.D. Black Holes and Entropy. Phys. Rev. D 1973, 7, 2333–2346. https://doi.org/10.1103/physrevd.7.2333.
- 2.
Bardeen, J.M.; Carter, B.; Hawking, S.W. The four laws of black hole mechanics. Commun. Math. Phys. 1973, 31, 161–170. https://doi.org/10.1007/bf01645742.
- 3.
Hawking, S.W. Particle creation by black holes. Commun. Math. Phys. 1975, 43, 199–220. https://doi.org/10.1007/bf02345020.
- 4.
Kastor, D.; Ray, S.; Traschen, J. Enthalpy and the mechanics of AdS black holes. Class. Quantum Gravity 2009, 26, 195011. https://doi.org/10.1088/0264-9381/26/19/195011.
- 5.
Dolan, B.P. Pressure and volume in the first law of black hole thermodynamics. Class. Quantum Gravity 2011, 28, 235017. https://doi.org/10.1088/0264-9381/28/23/235017.
- 6.
Kubizňák, D.; Mann, R.B. P–V criticality of charged AdS black holes. J. High Energy Phys. 2012, 2012, 33. https://doi.org/10.1007/jhep07(2012)033.
- 7.
Glavan, D.; Lin, C. Einstein–Gauss–Bonnet Gravity in Four‑Dimensional Spacetime. Phys. Rev. Lett. 2020, 124, 081301. https://doi.org/10.1103/physrevlett.124.081301.
- 8.
Fernandes, P.G.S.; Carrilho, P.; Clifton, T.; et al. The 4D Einstein–Gauss–Bonnet theory of gravity: A review. Class. Quantum Gravity 2022, 39, 063001. https://doi.org/10.1088/1361‑6382/ac500a.
- 9.
Liška, M.; Hennigar, R.A.; Kubizňák, D. No logarithmic corrections to entropy in shift‑symmetric Gauss–Bonnet gravity. J. High Energy Phys. 2023, 2023, 195. https://doi.org/10.1007/jhep11(2023)195.
- 10.
Kubizňák, D.; Liška, M. Can Bekenstein’s area law prevail in modified theories of gravity? Phys. Rev. D 2023, 108, L121501. https://doi.org/10.1103/physrevd.108.l121501.
- 11.
Wei, S.W.; Liu, Y.X. Extended thermodynamics and microstructures of four‑dimensional charged Gauss–Bonnet black hole in AdS space. Phys. Rev. D 2020, 101, 104018. https://doi.org/10.1103/physrevd.101.104018.
- 12.
Minamitsuji, M.; Maeda, K.I. Black hole thermodynamics in Horndeski theories. Phys. Rev. D 2023, 108, 084061. https://doi.org/10.1103/physrevd.108.084061.
- 13.
Alkac¸, G.; Özen, G.D.; Özşahin, H.; et al. Scaling symmetry, Smarr relation, and the extended first law in lower‑dimensional Lovelock gravity. Nucl. Phys. B 2024, 1002, 116535. https://doi.org/10.1016/j.nuclphysb.2024.116535.
- 14.
Yang, S.J.; Ali, M.S.; Wei, S.W.; et al. Holographic thermodynamics of a five‑dimensional neutral Gauss–Bonnet AdS black hole. Eur. Phys. J. C 2025, 85, 420. https://doi.org/10.1140/epjc/s10052‑025‑14136‑7.
- 15.
Lü, H.; Di Gennaro, S.; Ong, Y.C. Generalized entropy implies varying‑G: Horizon area dependent field equations and black hole‑cosmology coupling. Ann. Phys. 2025, 474, 169914. https://doi.org/10.1016/j.aop.2024.169914.
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