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

AdS/BCFT correspondence and BTZ black hole thermodynamics within Horndeski gravity