  We study the quantum gravitational corrections to the geometry of a
four-dimensionalcharged (Reissner-Nordstr\"om) 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.
