Static topological black hole with a nonminimal derivative coupling and a nonlinear electromagnetic field of Born-Infeld type

We consider scalar-tensor gravity with nonminimal derivative coupling and Born-Infeld electromagnetic field which is minimally coupled to gravity. Since the cosmological constant is taken into account it allowed us not only to derive a static black hole with a spherical horizon but also to obtain topological solutions with nonspherical horizons. The obtained metrics are thoroughly analyzed, namely for different distances and types of topology of horizon. To investigate singularities of the metrics Kretschmann scalar is used and it is shown that the character of singularity depends on the type of topology of horizon and dimension of space. We also investigate black hole's thermodynamics, namely we obtain and examine the black hole's temperature. To derive the first law of black hole's thermodynamics Wald's approach is applied. Nonetheless this approach is well established, there is ambiguity in definition of black hole's entropy which can be resolved just by virtue of some independent approach.