  We study the entropy relations of multi-horizons 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 to understanding the origin of black
hole entropy at the microscopic level. Revisiting the entropy product
introduced by Cvetic etc, 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 independence. When $Q$ vanishes in the
solution, it turns to mass dependence, even when including the effect of the
un-physical ``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
$\Lambda$ and the background topology $k$, and which does not depend on the
conserved charges $Q$, nor even the mass $M$.
