The entropy of dynamical black holes deviating from electrovacuum in second order

We study the entropy of dynamical black holes in Einstein–Maxwell theory. Starting from a stationary electrovacuum background with a bifurcate Killing horizon, the apparent horizon is constructed perturbatively in Gaussian null coordinates, and the area of its cross section is derived to second order. Within the covariant phase space formalism, we introduce a modified canonical energy that incorporates contributions from external matter and is naturally related to the second-order variation of the entropy through the balance law. Under the null energy condition and the asymptotic stationary condition, the entropy is proportional to the area of the apparent horizon up to second-order and satisfies the second law. Without the null energy condition, establishing this area law requires additional assumptions.