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