  We explore thermodynamic properties of a new class of Horndeski black holes
whose action contains a non-minimal kinetic coupling of a massless real scalar
and the Einstein tensor. Our treatment is based on the well-accepted
consideration, where the cosmological constant is dealt with as thermodynamic
pressure and the mass of black holes as thermodynamic enthalpy. We resort to a
newly introduced intensive thermodynamic variable, i.e., the coupling strength
of the scalar and tensor whose dimension is length square, and thus yield both
the generalized first law of thermodynamics and the generalized Smarr relation.
Our result indicates that this class of Horndeski black holes presents rich
thermodynamic behaviors and critical phenomena. Especially in the case of the
presence of an electric field, these black holes undergo two phase transitions.
Once the charge parameter exceeds its critical value, or the cosmological
parameter does not exceed its critical value, no phase transitions happen and
the black holes are stable. As a by-product, we point out that the coupling
strength acts as the thermodynamic pressure in the behavior of thermodynamics.
