We study the quantum gravitational corrections to the geometry of a four-dimensional charged (Reissner-Nordström) Anti de Sitter black hole starting from an effective field theory approach to quantum gravity. We use the expression of the modified horizon radius to compute the quantum corrected Wald entropy, whose expression reproduces the logarithmic behaviour found by other methods. We perform a thermodynamics analysis and compute the quantum gravitational corrections to the temperature, pressure, specific heat and Helmholtz free energy. All these quantities are renormalisation group invariant. We find that a quantum charged AdS black hole can exist only for a bounded range of masses and that it can undergo a second order phase transition as it moves from a state with positive specific heat to a negative one.

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Starting from an effective action for quantum gravity, we calculate the quantum gravitational corrections to the Wald entropy of a four dimensional non-extremal Reissner–Nordström (RN) black hole in the limit of small electric charge, generalising a previous calculation carried out by Calmet and Kuipers (Phys Rev D 104(6):066012, 2021) for a Schwarzschild black hole. We show that, at second order in the Ricci curvature, the RN metric receives quantum corrections which shift the classical position of the event horizon. We apply the Wald entropy formula by integrating over the perimeter of the quantum corrected event horizon. We then compute the quantum gravitational corrections to the temperature and the pressure of the black hole.

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