For any entropy function S(r_h), the Goon-Penco extremality ratio equals d(pi r_h^2)/dS, so the original relation survives only for the area law.
PV criticality of the second order quantum corrected Horava-Lifshitz black hole
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In this paper, higher order quantum gravity effects on the thermodynamics of Ho\v{r}ava-Lifshitz black hole investigated. Both Kehagius-Sfetsos and Lu-Mei-Pop solutions of Ho\v{r}ava-Lifshitz black hole considered and higher order corrected thermodynamics quantities obtained. The first order correction is logarithmic and second order correction considered proportional to the inverse of entropy. These corrections are due to the thermal fluctuation and interpreted as quantum loop corrections. Effect of such quantum corrections on the stability and critical points of Horava-Lifshitz black holes studied. We find that higher order correction affects critical point and stability of Lu-Mei-Pop solution and yield to the second order phase transition for the case of Kehagius-Sfetsos solution.
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Quantum Corrections and Extremality: A Generalized Universal Relation
For any entropy function S(r_h), the Goon-Penco extremality ratio equals d(pi r_h^2)/dS, so the original relation survives only for the area law.