Geometric model of black hole quantum $N$-portrait, extradimensions and   thermodynamics
  Recently a short scale modified black hole metric, known as holographic metric, has been proposed in order to capture the self-complete character of gravity. In this paper we show that such a metric can reproduce some geometric features expected from the quantum $N$-portrait beyond the semi-classical limit. We show that for a generic $N$ this corresponds to having an effective energy momentum tensor in Einstein equations or, equivalently, non-local terms in the gravity action. We also consider the higher dimensional extension of the metric and the case of an AdS cosmological term. We provide a detailed thermodynamic analysis of both cases, with particular reference to the repercussions on the Hawking-Page phase transition. 
Antonia M. Frassino, Sven K\"oppel and Piero Nicolini
gr-qc hep-th