Hagedorn versus Hawking-Page transition in string theory

We study the supergravity dual to the confinement or deconfinement phase transition for the N=4 SU(N) super Yang-Mills theory on R×S³ with a chemical potential conjugate to a U(1)⊂SO(6)R charge. The appropriate supergravity system is a single-charge black hole in $D=5$ N=8 gauged supergravity. The application of the gauge-string theory holographic renormalization approach leads to new expressions for the black hole Arnowitt-Deser-Misner mass and its generalized free energy. We comment on the relation of this phase transition to the Hagedorn transition for strings in the maximally supersymmetric plane wave background with null Ramond-Ramond five-form field strength.