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Gravity is believed to have deep and inherent relation to thermodynamics. Westudy phase transition and critical behavior in the extended phase space ofasymptotic anti de-Sitter (AdS) black holes in Einstein-Horndeski gravity. Wedemonstrate that the black hole in Einstein-Horndeski gravity undergo phasetransition and P-V criticality mimicking the van der Waals gas-liquid system.The key approach in our study is to introduce a more reasonable pressureinstead of previous pressure $P=-\Lambda/8\pi$ related to cosmological constant$\Lambda$, and this proper pressure is given insight from the asymptoticalbehaviour of this black hole. Moreover, we also first obtain P-V criticality inthe two cases with $\Lambda=0$ and $\Lambda>0$ in our paper, which implicatesthat the cosmological constant $\Lambda$ may be not a necessary pressurecandidate for black holes at the microscopic level. We present criticalexponents for these phase transition processes.

P-V criticality in the extended phase space of black holes in Einstein-Horndeski gravity