  We re-derive the first law of black hole mechanics in the context of the
Einstein-Maxwell theory in a gauge-invariant way introducing "momentum maps"
associated to field strengths and the vectors that generate their symmetries.
These objects play the role of generalized thermodynamical potentials in the
first law and satisfy generalized zeroth laws, as first observed in the context
of principal gauge bundles by Prabhu, but they can be generalized to more
complex situations. We test our ideas on the $d$-dimensional
Reissner-Nordstr\"om-Tangherlini black hole.
