Conserved charges and the first law of black holes for field dependent symmetry generators in the covariant phase space formalism

In this paper, we generalize the covariant phase space formalism conjugate to field dependent vectors to include internal gauge transformations when gauge fields are present. In our formalism, the symmetry generators are combination of diffeomorphisms plus internal gauge transformations and depend on the field configuration. When the field dependence of the symmetry generators is considered in the covariant phase space formalism, the first law of black hole thermodynamics is a direct result for generally invariant gravitational theories. To check the validity of our formalism, we investigate the conserved charges and the first laws of thermodynamics for a torus-like black hole in Einstein-Maxwell theory and a charged Einstein-Euler-Heisenberg AdS black hole in Einstein-Euler-Heisenberg nonlinear electrodynamics.