holographic dual picture of a modified horndeski black hole

The usual horndeski black hole do not have $ p-v $ critical points, hence do not show any phase transitions. in this article we a particular modified horndenski black hole is considered to study the $ p-v $ diagram and the phase transitions. we show that this modified horndeski black hole solution satisfies the textit{ist} order phase transition. we also show that the modified horndeski black hole is holographic dual of a textit{van der waals}(vdw) fluid. finally, we study the thermodynamics of modified horndeski black hole based on the equation of state originating from the slope of temperature versus entropy. this new prescription provides us a simple and powerful way to study the critical behavior and the phase transition of black holes. the analytical interpretation of possible phase transition points leads us to set some nonphysical range on the horizon radius for the black hole.