First order flow for non-extremal AdS black holes and mass from holographic renormalization

A bstract In this paper we present a first order formulation for non-extremal Anti-de Sitter black hole solutions in four dimensional N =2 U(1)-gauged Supergravity. The dynamics is determined in terms of a quantity W which plays the role of a superpotential for the gauging potential in the action. We show how the first order flow arises from writing the action as a sum of squares and we identify the superpotential driving the first order flow for two classes of solutions (electric and magnetic) of the t 3 model. After identifying W , we study the Hamilton-Jacobi holographic renormalization procedure in presence of mixed boundary conditions for the scalar fields. We compute the renormalized on-shell action and the mass of the black hole configurations. The expression obtained for the mass satisfies the first law of thermodynamics.