Static topological dilatonic black hole with multiple horizons and its thermodynamics
We obtain a static topological charged black hole solution within Einstein-dilaton theory with nonlinear electromagnetic field represented by an infinite series over Maxwell field invariant. In contrast with standard Einstein-Maxwell-dilaton (EMD) theory where the black hole might have no more than two horizons, the nonlinear generalization might give rise to a solution with arbitrary number of horizons. We examine energy conditions and show that for the nonlinear electromagnetic field itself all the energy conditions are fulfilled outside of the black hole, whereas for the dilaton field the only condition which is not violated outside is the Null Energy Condition (NEC). We also study thermodynamics of the black hole and show that the thermal behaviour of the solution is considerably richer than for its standard EMD cousin. To have complete description of the thermodynamics we also use the Euclidean method, which allows us to consider the so-called Grand Canonical and Canonical Ensembles. The results obtained by different approaches show their consistency. The Euclidean description naturally allowed us to consider global stability, which is shown to have some common features with standard EMD solution, or even Reissner-Nordstr\"{o}m black hole. But there some new peculiarities caused by the nonlinear field, namely we might have additional stable phases and a triple point. To examine near critical behaviour we consider the extended phase space approach, where the cosmological constant is supposed to be a thermodynamic variable. Making use of the extended formalism we also obtain the Smarr relation.