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In this paper, we obtain higher dimensional topological black hole solutionsof Einstein-$\Lambda$ gravity in the presence of a class of nonlinearelectrodynamics. First, we calculate the conserved and thermodynamic quantitiesof ($n+1$)-dimensional asymptotically flat solutions and show that they satisfythe first law of thermodynamics. Also, we investigate the stability of thesesolutions in the (grand) canonical ensemble. Second, we endow a global rotationto the static Ricci-flat solutions and calculate the conserved quantities ofsolutions by using the counterterm method. We obtain a Smarr-type formula forthe mass as a function of the entropy, the angular momenta and the electriccharge, and show that these quantities satisfy the first law of thermodynamics.Then, we perform a stability analysis of the rotating solutions both in thecanonical and the grand canonical ensembles.

Thermodynamic instability of topological black holes with nonlinear source