Generalized First Law and Smarr Formula: Beyond Additivity and Extensivity

Abstract: The study of black hole thermodynamics becomes a central topic in gravitational physics, where the first law and the Smarr relation establish a deep connection between spacetime geometry and thermodynamic laws. As we know, these relations depend on the entropy; any modification to the entropy arising from quantum gravity or generalized statistical mechanics may impact the basic thermodynamic framework of black holes. In this work, we develop a general framework for deriving the first law of black hole thermodynamics and the associated Smarr relation for generic spherically symmetric spacetime under a wide class of generalized entropy models. In addition, a generalized Ruppeiner thermodynamic geometry is developed to utilize the generalized entropy model, from which the curvature scalar is determined in a general form. To demonstrate this framework, we assume the Resinser-Nordström black hole and investigate the corresponding extremal and non-extremal phase transition. Interestingly, our analysis reveals that entropy models consistent with the Abè-type composition rule result in a vanishing thermodynamic curvature, whereas violations of this rule exhibit curvature divergences, suggesting a geometric test for the consistency of generalized entropy models.