Instabilities and doubly spinning black holes

Ultra-spinning black holes in higher dimensions have qualitatively similar thermodynamical properties to the black strings/p-branes- they exhibit a black membrane-like phase. In these notes we will explore which doubly spinning black holes and black rings show this black membrane-like phase and pinpoint the threshold of these regimes. For the same cases we will study the thermodynamical instabilities in the grand canonical ensemble and compute the zeros of the Hessian of the Gibbs potential. Comparing our results, we will show that the thresholds of the black membrane phase and the thermal instability do not necessarily coincide.