phase space in which the cosmological constant, nonlinearity and Lovelock parameters considered as dynamical variables. We have calculated generalized Smarr formula and also modified first law of thermodynamics. Extended phase space help us to investigate the similarities between the thermodynamical behavior of black hole system under studied and the Van der Waals gas/liquid system.

Finally, we should note that for the sake of economy, we investigated stability conditions only for $k=1$. One may regard other horizon topology for discussion of thermal stability. In addition, it is worthwhile to mention that it would be interesting to investigate the phase transition by Geometrothermodynamics approach [@GTD]. In addition, one can follow the section IV to discuss about the concept of extended phase space thermodynamics and $P-V$ criticality of the Lovelock black holes with BI type NED [@PVlovelock]. We leave these problems to our forthcoming independent works.

We would like to thank the anonymous referee for useful suggestions and enlightening comments. The authors wish to thank Shiraz University Research Council. This work has been supported financially by Center for Excellence in Astronomy & Astrophysics of Iran (CEAAI-RIAAM).

**Appendix**

The action of third order Lovelock gravity in the presence of NED which is related to 