  In this work, we study gravity duals of conformal field theories with
boundaries known as AdS/BCFT correspondence, put forward by Takayanagi [1],
within a scalar-tensor theory, as proposed by Horndeski in 4D [2]. We consider
the case of 3D gravity dual to 2D BCFT, take a Gibbons-Hawking surface term
modified by Horndeski's theory and find the corresponding 3D (BTZ) black hole
solutions. We analyze the effects of Horndeski's gravity on the profile of the
extra boundary for the black hole by using an approximate analytic solution.
Performing a holographic renormalization, we have calculated the free energy
and obtained the total entropy and the corresponding area, as well as the
boundary entropy for the black hole. In particular, the boundary entropy found
here, can be seen as an extension of the one proposed by Takayanagi. From the
free energy, we have done a systematic study of the 3D black hole
thermodynamics and present, among our results, an indication of restoration of
conformal symmetry for high temperatures. Finally, we present a study on the
influence of the Horndeski's gravity on the Hawking-Page phase transition where
we can see the stable and unstable phases throughout the plane of free energy
{\sl versus} temperature.
