  Black holes are famous for their universal behavior. New thermodynamic
relations have been found recently for the product of gravitational entropies
over all the horizons of a given stationary black hole. This product has been
found to be independent of the mass for all such solutions of Einstein-Maxwell
theory in d=4,5. We study the universality of this mass independence by
introducing a number of possible higher curvature corrections to the
gravitational action. We consider finite temperature black holes with both
asymptotically flat and (A)dS boundary conditions. Although we find examples
for which mass independence of the horizon entropy product continues to hold,
we show that the universality of this property fails in general. We also derive
further thermodynamic properties of inner horizons, such as the first law and
Smarr relation, in the higher curvature theories under consideration, as well
as a set of relations between thermodynamic potentials on the inner and outer
horizons that follow from the horizon entropy product, whether or not it is
mass independent.
