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.