Universal properties and the first law of black hole inner mechanics

We show by explicit computations that the product of all the horizon areas is independent of the mass, regardless of the topology of the horizons. The universal character of this relation holds for all known five-dimensional asymptotically flat black rings, and for black strings. This gives further evidence for the crucial role that the thermodynamic properties at each horizon play in understanding the entropy at the microscopic level. To this end we propose a first law for the inner Cauchy horizons of black holes. The validity of this formula, which seems to be universal, was explicitly checked in all cases.