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