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

In this paper we present a first order formulation for non-extremal Anti-de Sitter black hole solutions in four dimensional $\mathcal{N}=2$ U(1)-gauged Supergravity. The dynamics is determined in terms of a quantity $\mathcal{W}$ 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 $\mathcal{W}$, 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.