The instability of low-temperature black holes in gauged $$ \mathcal{N} $$ = 8 supergravity

We consider the static planar black hole solutions in the STU model of the gauged N = 8 supergravity in four dimensions. We give a straightforward derivation of the equation of state of the purely electric and purely magnetic solutions with four charges. Then we give a simple proof that the determinant of the Hessian of the energy is always negative below some critical finite temperature for the purely electric solutions. We compute the spinodal line for the usual planar Reissner-Nordström solution in four dimensions. Inspired by the magnetic superalgebra we show that the supersymmetric solutions are metastable if the energy is restricted to satisfy the topological twist condition ab initio and it is shifted to be zero on the BPS solutions.