  We derive the first law of black hole mechanics for physical theories based
on a local, covariant and gauge-invariant Lagrangian where the dynamical fields
transform non-trivially under the action of internal gauge transformations. The
theories of interest include General Relativity formulated in terms of tetrads,
Einstein-Yang-Mills theory and Einstein-Dirac theory. Since the dynamical
fields of these theories have gauge freedom, we argue that there is no group
action of diffeomorphisms of spacetime on such dynamical fields. In general,
such fields cannot even be represented as smooth, globally well-defined tensor
fields on spacetime. Consequently the derivation of the first law by Iyer-Wald
cannot be used directly. We show how such theories can be formulated on a
principal bundle and that there is a natural action of automorphisms of the
bundle on the fields. These bundle automorphisms encode both spacetime
diffeomorphisms and gauge transformations. Using this reformulation we define
the Noether charge associated to an infinitesimal automorphism and the
corresponding notion of stationarity and axisymmetry of the dynamical fields.
We define certain potentials and charges at the horizon of a black hole so that
the potentials are constant on the bifurcate Killing horizon, giving a
generalised zeroth law for bifurcate Killing horizons. We identify the
gravitational potential and perturbed charge as the temperature and perturbed
entropy of the black hole which gives an explicit formula for the perturbed
entropy analogous to the Wald entropy formula. We obtain a general first law of
black hole mechanics for such theories. The first law relates the perturbed
Hamiltonians at spatial infinity and the horizon, and the horizon contributions
take the form of a potential times perturbed charge term. We also comment on
the ambiguities in defining a prescription for the total entropy for black
holes.
