First law of black hole mechanics in variable background fields

It is well known that in general theories of gravity with the diffeomorphism symmetry, the black hole entropy is a Noether charge. But what will happen if the symmetry is explicitly broken? By investigating the covariant first law of black hole mechanics with background fields, we show that the would-be Noether charge still can be identified as the black hole entropy, provided that it is a local quantity on the horizon. Moreover, motivated by the proposal that the cosmological constant behaves as a thermodynamic variable, we allow the non-dynamical background fields to be varied. To illustrate this general formalism, we study a generic static black brane in the massive gravity. Using the first law and the scaling argument, we obtain two Smarr formulas. We show that both of them can be retrieved without relying on the first law, hence providing a self-consistent check of the theory.