Spinning black holes for generalized scalar-tensor theories in three dimensions

We consider a general class of scalar tensor theories in three dimensions whose action contains up to second-order derivatives of the scalar field with coupling functions that only depend on the standard kinetic term of the scalar field, thus ensuring the invariance under the constant shift of the scalar field. For this model, we show that the field equations for a stationary metric ansatz together with a purely radial scalar field can be fully integrated. The kinetic term of the scalar field solution is shownto satisfy an algebraic relation depending only on the coupling functions, and hence is constant while the metric solution is nothing but a Banados-Teitelboim-Zanelli (BTZ)-like metric with an effective cosmological constant fixed in terms of the coupling functions. As a direct consequence the thermodynamics of the solution is shown to be identical to a BTZ-like one with an effective cosmological constant, despite the presence of a scalar field. Finally, the expression of the semiclassical entropy of this solution is also confirmed through a generalized Cardy-like formula involving the mass of the scalar soliton obtained from the black hole by means of a double Wick rotation.