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 Fluctuations of a BTZ Black Hole in Massive Gravity
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In this work, we shall analyze the effects of quantum fluctuations on the properties of a BTZ black hole, in a massive theory of gravity. We will analyze this for a charged BTZ black hole in asymptotically AdS and dS space-times. The quantum fluctuations would produce thermal fluctuations in the thermodynamics of this BTZ black hole. As these fluctuations would become relevant at a sufficiently small scale, we shall discuss the effects of such thermal fluctuations on the entropy of a small charged BTZ black. We shall also analyze the effects of these fluctuations on the stability of such a black hole.
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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.