entanglement entropy in horndeski gravity

In this work, we explore the holographic entanglement entropy with an infinite strip region of the boundary in horndeski gravity. in our prescription, we consider the spherical and planar topologies black holes in the ads4/cft3 scenario. in such a framework, we show the behavior of the entanglement entropy in the function of the horndeski parameters. such parameters modify the information store of subsystem a, especially when the parameter γ increases the information about the subsystem will also increase or decrease when it decreases. thus, with this scheme we compute the first law of entanglement thermodynamics in horndeski gravity and we show that a very small subsystem obeys the analogous property of the first law of thermodynamics if we excite the system.