  In this paper, we study Einstein gravity either minimally or non-minimally
coupled to a vector field which breaks the gauge symmetry explicitly in general
dimensions. We first consider a minimal theory which is simply the
Einstein-Proca theory extended with a quartic self-interaction term for the
vector field. We obtain its general static maximally symmetric black hole
solution and study the thermodynamics using Wald formalism. The aspects of the
solution are much like a Reissner-Nordstr{\o}m black hole in spite of that a
global charge cannot be defined for the vector. For non-minimal theories, we
obtain a lot of exact black hole solutions, depending on the parameters of the
theories. In particular, many of the solutions are general static and have
maximal symmetry. However, there are some subtleties and ambiguities in the
derivation of the first laws because the existence of an algebraic degree of
freedom of the vector in general invalids the Wald entropy formula. The
thermodynamics of these solutions deserves further studies.
