Thermodynamic relations for entropy and temperature of multi-horizons black holes

Abstract

We present some entropy and temperature relations of multi-horizons, even including the "virtual" horizon. These relations are related to product, division and sum of entropy and temperature of multi-horizons. We obtain the additional thermodynamic relations of both static and rotating black holes in three and four dimensional (A)dS spacetime. Especially, a new dimensionless, charges-independence and $T_+S_+=T_-S_-$ like relation is presented. This relation does not depend on the mass, electric charge, angular momentum and cosmological constant, as it is always a constant. These relations lead us to get some interesting thermodynamic bound of entropy and temperature, including the Penrose inequality which is the first geometrical inequality of black holes. Besides, based on these new relations, one can obtain the first law of thermodynamics and Smarr relation for all horizons of black hole.