Effective thermodynamical system of Schwarzschild�de Sitter black holes from R�nyi statistics

Abstract

It has been known that the Schwarzschild--de Sitter (Sch-dS) black hole may not be in thermal equilibrium and also be found to be thermodynamically unstable in the standard black hole thermodynamics. In the present work, we investigate the possibility to realize the thermodynamical stability of the Sch-dS black hole as an effective system by using the R\'enyi statistics, which includes the nonextensive nature of black holes. Our results indicate that the nonextensivity allows the black hole to be thermodynamically stable, which gives rise to the lower bound on the nonextensive parameter. By comparing the results to ones in the separated system approach, we find that the effective temperature is always smaller than the black hole horizon temperature and the thermodynamically stable black hole in the effective approach is always larger than the one in the separated approach at a certain temperature. There exists only the zeroth-order phase transition from the hot gas phase to the black hole phase for the effective system, while it is possible to have transitions of both zeroth order and first order for the separated system.