  By regarding the Einstein equations as equation(s) of state, we demonstrate
that a full cohomogeneity horizon first law can be derived in horizon
thermodynamics. In this approach both the entropy and the free energy are
derived concepts, while the standard (degenerate) horizon first law is
recovered by a Legendre projection from the more general one we derive. These
results readily generalize to higher curvature gravities and establish a way of
how to formulate consistent black hole thermodynamics without conserved
charges.
