  We study entanglement entropy in a particular tensor-scalar theory: Horndeski
gravity. Our goal is two-fold: investigate the Lewkowycz-Maldacena proposal for
entanglement entropy in the presence of a tensor-scalar coupling and address a
puzzle existing in the literature regarding the thermal entropy of
asymptotically AdS Horndeski black holes. Using the squashed cone method, i.e.
turning on a conical singularity in the bulk, we derive the functional for
entanglement entropy in Horndeski gravity. We analyze the divergence structure
of the bulk equation of motion. Demanding that the leading divergence of the
transverse component of the equation of motion vanishes we identify the surface
where to evaluate the entanglement functional. We show that the surface
obtained is precisely the one that minimizes said functional. By evaluating the
entanglement entropy functional on the horizon we obtain the thermal entropy
for Horndeski black holes; this result clarifies discrepancies in the
literature. As an application of the functional derived we find the minimal
surfaces numerically and study the entanglement plateaux.
