  In this paper, we investigate the critical points of the $ P-V $ diagram and
the phase transitions of the Horndenski black holes. In fact, the usual
Horndeski black holes do not have $ P-V $ critical points, hence do not show
any phase transitions. However, we successfully modify the Horndeski black hole
solution to obtain such a phase transition behavior. This modified black holes
solution is satisfied by the equation of state of liquid-gas phase transition.
Also, we study the thermodynamics of our modified Horndeski black hole by
applying a new method based on the equation of state that originated from the
slope of temperature versus entropy. This new prescription provides us a simple
powerful way to study the critical behavior and the phase transition of the
black holes and concludes the novel results. The analytical interpretation of
possible phase transition points leads us to set some nonphysical range on the
horizon radius for the black hole.
