  We consider scalar-tensor gravity with nonminimal derivative coupling and
Born-Infeld electromagnetic field which is minimally coupled to gravity. Since
cosmological constant is taken into account it allowed us not only derive
static black hole with spherical horizon but also to obtain topological
solutions with non-spherical 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 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.
