Generalized First Law and Smarr-type Relation: Beyond Additivity and Extensivity

Abstract

In this work, we develop a general framework for deriving the first law of black hole thermodynamics and the associated Smarr-type relation for 4-dimensional static spherically symmetric spacetimes within a broad class of generalized entropy models. In addition, by employing the entropy representation for the Reissner--Nordström (RN) black hole, we determine a concise curvature functional that depends only on this representation and its derivatives. A general flatness condition is derived, demonstrating that when the entropy Hessian metric is regular and nondegenerate, both homogeneous power-law transformations and a separate logarithmic branch leave the RN black hole's Ruppeiner geometry flat. On the other hand, Rényi, Sharma-Mittal, Kaniadakis, and LQG-inspired entropy deformations typically give rise to nonvanishing curvature in the regular thermodynamic regime, demonstrating that the extremality of the RN black hole does not by itself require a curvature divergence.

arXiv comment14 pages, title changed, further investigations added