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After considering the quantum corrections of Einstein-Maxwell theory, theeffective theory will contain some higher-curvature terms and nonminimallycoupled electromagnetic fields. In this paper, we study the first law of blackholes in the gravitational electromagnetic system with the Lagrangian$\math{L}(g_{ab}, R_{abcd}, F_{ab})$. Firstly, we calculate the Noether chargeand the variational identity in this theory, and then generically derive thefirst law of thermodynamics for an asymptotically flat stationaryaxisymmetrical symmetric black hole without the requirement that theelectromagnetic field is smooth on the bifurcation surface. Our resultsindicate that the first law of black hole thermodynamics might be valid for theEinstein-Maxwell theory with some quantum corrections in the effective region.

First Law of Black Hole in the Gravitational Electromagnetic System