Dynamical entropy of charged black objects

We develop a general framework for electromagnetic potential-charge contributions to the first law of black hole mechanics, applicable to dynamical first-order perturbations of stationary black objects with possibly non-compact bifurcate Killing horizons. Working in the covariant phase space formalism, we derive both comparison and physical process versions of the first law. We consider generic diffeomorphism-invariant theories of gravity in $D$ spacetime dimensions, containing non-minimally coupled abelian $p$-form gauge fields. The pullback of the gauge field to the horizon is allowed to diverge while its field strength remains smooth, yielding gauge-invariant electric potential-charge pairs in the first law. We further extend the construction to include magnetic charges by developing a bundle-covariant, gauge-invariant prescription that fixes the Jacobson-Kang-Myers ambiguity in the improved Noether charge. Electric and magnetic charges are, respectively, associated with non-trivial $(D - p - 1)$- and $(p + 1)$-cycles of the horizon cross-section, whose homology classes determine the number of independent potential-charge pairs through the Betti numbers $b_{D - p - 1}$ and $b_{p + 1}$. Further, the dynamical gravitational entropy entering the first law is identified with the gauge-invariant part of the improved Noether charge, giving a gauge-invariant extension of the recent proposal by Hollands, Wald and Zhang. We illustrate our framework with dyonic AdS black holes, dipole black rings, and charged black branes.