Topological black hole in the theory with nonminimal derivative coupling with power-law Maxwell field and its thermodynamics

We obtain topological black hole solutions in scalar-tensor gravity with nonminimal derivative coupling between scalar and tensor components of gravity and a power-law Maxwell field minimally coupled to gravity. The obtained solutions can be treated as a generalization of previously derived charged solutions with standard Maxwell action [X.-H. Feng, H.-S. Liu, H. Lu, and C. N. Pope, Phys. Rev. D 93, 044030 (2016)]. We examine the behavior of obtained metric functions for some asymptotic values of distance and coupling. To obtain information about singularities of the metrics we calculate Kretschmann scalar. We also examine the behavior of gauge potential and show that it is necessary to impose some constraints on parameter of nonlinearity in order to obtain reasonable behavior of the filed. The next part of our work is devoted to the examination of black hole thermodynamics. Namely, we obtain the black hole's temperature and investigate it in general as well as in some particular cases. To introduce entropy, we use the well-known Wald procedure which can be applied to quite general diffeomorphism-invariant theories. We also extend thermodynamic phase space by introducing thermodynamic pressure related to cosmological constant and as a result we derive generalized first law and Smarr relation. The extended thermodynamic variables also allow us to construct Gibbs free energy and its examination gives important information about thermodynamic stability and phase transitions. We also calculate heat capacity of the black holes which demonstrates variety of behavior for different values of allowed parameters.