---
abstract: 'In this paper, we obtain topological black hole solutions of third order Lovelock gravity couple with two classes of Born-Infeld type nonlinear electrodynamics with anti-de Sitter asymptotic structure. We investigate geometric and thermodynamics properties of the solutions and obtain conserved quantities of the black holes. We examine the first law of thermodynamics and find that the conserved and thermodynamic quantities of the black hole solutions satisfy the first law of thermodynamics. Finally, we calculate the heat capacity and determinant of Hessian matrix to evaluate thermal stability in both canonical and grand canonical ensembles. Moreover, we consider extended phase space thermodynamics to obtain generalized first law of thermodynamics as well as extended Smarr formula.'
author:
- 'S. H. Hendi$^{1,2}$[^1] and A. Dehghani$^1$'
title: |
    Thermodynamics of third order Lovelock adS black holes\
    in the presence of Born-Infeld type nonlinear electrodynamics
---

Introduction
============

Regarding string theory and brane world cosmology, it has been shown that spacetimes possess more than four dimensions. Taking into account higher dimensional spacetimes, we know that the general conserved symmetric tensor that depends on the metric and its derivatives up to second order is not the Einstein tensor.

One of the natural generalization of Einstein 