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.
Generalized second law of thermodynamics in black hole physics,
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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.