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.
Quantum gravity effects on Ho\v{r}ava-Lifshitz black hole
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In this paper, we would like to obtain quantum gravity effects by using Ho\v{r}ava-Lifshitz black hole. We consider logarithmic corrected thermodynamics quantities and investigate the effects of logarithmic correction term. Logarithmic correction comes from thermal fluctuation and may be interpreted as quantum loop corrections. As black hole is a gravitational system, hence we can investigate quantum gravity effect. We find such effects on the black hole stability and obtain domain of correction coefficient.
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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.