Slowly rotating black hole solution in the scalar-tensor theory with nonminimal derivative coupling and its thermodynamics

We obtain a slowly rotating black hole solution in the scalar-tensor theory of gravity with nonminimal derivative coupling to the Einstein tensor. Properties of the obtained solution have been examined carefully. We also investigate the thermodynamics of the given black hole. To obtain thermodynamic functions, namely its entropy we use the Wald procedure which is suitable for quite general diffeomorphism-invariant theories. The applied approach allowed us to obtain the expression for entropy and the first law of black hole thermodynamics. Having introduced thermodynamic pressure which is related to the cosmological constant we have examined thermodynamics of the black hole in the so called extended phase space. The extended phase space and specifically chosen scalar `charge' allowed us not only to obtain the generalized first law but also derive the Smarr relation. The behaviour of black hole's temperature, heat capacity and Gibbs free energy shows a lot of similarities with the behaviour of the corresponding values for Schwarzschild-AdS black hole in standard General Relativity.