thermodynamic properties.

The outline of the paper is as follows. We present the topological black hole solutions in Sec. \[Topol\]. Sec. \[Thermodynamics\], is devoted to investigate conserved and thermodynamic quantities of topological black holes. We also analyze the thermodynamic stability of the solutions in the canonical and grand canonical ensembles. We finish our paper with some conclusions.

Topological adS black holes in third order Lovelock gravity [Topol]{}
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The gravitational and electromagnetic field equations of third order Lovelock gravity in the presence of NED may be written as $$G_{\mu \nu }+\Lambda g_{\mu \nu }+\alpha _{2}\mathcal{G}_{\mu \nu }+\alpha
_{3}\mathcal{H}_{\mu \nu }=\frac{1}{2}g_{\mu \nu }\mathcal{L}(\mathcal{F}%
)-2F_{\mu \lambda }F_{\nu }^{\;\lambda }\mathcal{L}_{\mathcal{F}},
\label{Geq}$$ $$\partial _{\mu }\left( \sqrt{-g}\mathcal{L}_{\mathcal{F}}F^{\mu \nu }\right)
=0,  \label{BIeq}$$where $G_{\mu \nu }$ is the Einstein tensor, $\mathcal{G}_{\mu \nu }$ and $%
\mathcal{H}_{\mu \nu }$ are, respectively, the second and third orders Lovelock tensor given as $$\mathcal{G}_{\mu \nu } =2(R_{\mu \sigma \kappa \tau }R_{\nu }^{\phantom{\nu}%
\sigma \kappa \tau }-2R_{\mu \rho \nu \sigma }R^{\rho \sigma }-2R_{\mu
\sigma }R_{\phantom{\sigma}\nu }^{\sigma }+RR_{\mu \nu })-\frac{1}{2}g_{\mu
\nu }\mathcal{L}_{2},  \label{Love2}$$ $$\begin{aligned}
\mathcal{H}_{\mu \nu } &=&-3[4R^{\tau \rho \sigma \kappa }R_{\sigma \kappa
\lambda \rho }R_{\phantom{\lambda }{\nu \tau \mu}}^{\lambda }-8R_{%
\phantom{\tau \rho}{\lambda \sigma}}^{\tau \rho }R_{\phantom{\sigma
\kappa}{\tau \mu}}^{\sigma \kappa }R_{\phantom{\lambda }{\nu \rho \kappa}%
}^{\lambda 