Additivity of the entropies of black holes and matter in equilibrium

We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding matter. The grand canonical ensemble is defined by the geometry of the boundary of a region of spacetime and the chemical potential evaluated on this boundary. Additivity of the actions of gravity and matter and the principle of equivalence then jointly imply that the total entropy is the simple sum of the accepted black-hole entropy and the ordinary matter entropy. However, energy and thermodynamic potentials exhibit interaction terms and are not simply additive