The Entropy Sum of (A)dS Black Holes in Four and Higher Dimensions

Abstract

We present the "entropy sum" relation of (A)dS charged black holes in higher dimensional Einstein-Maxwell gravity, $f(R)$ gravity, Gauss-Bonnet gravity and gauged supergravity. For their "entropy sum" with the necessary effect of the un-physical "virtual" horizon included, we conclude the general results that the cosmological constant dependence and Gauss-Bonnet coupling constant dependence do hold in both the four and six dimensions, while the "entropy sum" is always vanishing in odd dimensions. Furthermore, the "entropy sum" of all horizons is related to the geometry of the horizons in four and six dimensions. In these explicitly four cases, one also finds that the conserved charges $M$ (the mass), $Q$ (the charge from Maxwell field or supergravity) and the parameter $a$ (the angular momentum) play no role in the "entropy sum" relations.