AdS dyonic black hole and its thermodynamics

We obtain spherically-symmetric and R-2-symmetric dyonic black holes that are asymptotic to anti-de Sitter space-time (AdS), which are solutions in maximal gauged four-dimensional supergravity, with just one of the U(1) fields carrying both the electric and magnetic charges (Q, P). We study the thermodynamics, and find that the usually-expected first law does not hold unless P = 0, Q = 0 or P = Q. For general values of the charges, we find that the first law requires a modification with a new pair of thermodynamic conjugate variables. We show that they describe the scalar hair that breaks some of the asymptotic AdS symmetries.