Entropy relations of black holes with multihorizons in higher dimensions

We study the entropy relations of multihorizon black holes in higher dimensional (A)dS spacetime with maximal symmetries, including Einstein-Maxwell gravity and f(R)(-Maxwell) gravity. These additional equalities in thermodynamics are expected to be useful in understanding the origin of black hole entropy at the microscopic level. By revisiting the entropy product introduced by Cvetic et al., we see that in our case it has an unexpected behavior. It is shown that this electric charge Q plays an important role in this entropy product. The entropy product of charged black holes only depends on the electric charge Q and is mass independent. When Q vanishes in the solution, it becomes mass dependent, even when including the effect of the unphysical "virtual" horizons. In this sense, the "universal relation" of this entropy product is destroyed. Then we introduce another kind of "universal" entropy relation, which only depends on the cosmological constant. and the background topology k and does not depend on the conserved charges Q or even the mass M.