does not affect the temperature, entropy and finite mass only for black holes with Ricci flat horizon, $k=0$. Moreover, we showed that the thermodynamics and conserved quantities satisfy the first law of thermodynamics.

We performed stability criterion in both canonical and grand canonical ensembles by use of numerical analysis only for $k=1$. We found a lower bound for the horizon radius, $r_{0}\geq 0$, in which the temperature is positive for $r_{+}>r_{0}$. We showed that the nonlinearity parameter, $%
\beta $, and also Lovelock coefficient, $\alpha $ can affect the value of $%
r_{0}$. Then we studied the heat capacity and determinant of Hessian matrix and showed that for $\alpha <\alpha _{c}$, there are two limits $r_{a}$ and $%
r_{b}$ ($r_{a}<r_{b}$), in which the black holes have an unstable phase for $%
r_{a}<r_{+}<r_{b}$. Furthermore, we found an lower limit ($r_{c}$) in which for $\alpha >\alpha _{c}$ and $r_{+}>r_{c}$ the black holes are stable. In addition, we found that for canonical ensemble one finds $r_{c}=r_{0}$ and for grand canonical ensemble $r_{c}>r_{0}$. Calculations showed that regardless of value of values of $\beta $, there is an unstable phase of black hole solutions for $\alpha <\alpha _{c}$.

At last, we have discussed the extended 