Thermodynamical properties of topological Born-Infeld-dilaton black holes

We examine the $(n+1)$-dimensional $(n\ge 3)$ action in which gravity is coupled to the Born-Infeld nonlinear electrodynamic and a dilaton field. We construct a new $(n+1)$-dimensional analytic solution of this theory in the presence of Liouville-type dilaton potentials. These solutions which describe charged topological dilaton black holes with nonlinear electrodynamics, have unusual asymptotics. They are neither asymptotically flat nor (anti)-de Sitter. The event horizons of these black holes can be an $(n-1)$-dimensional positive, zero or negative constant curvature hypersurface. We also analyze thermodynamics and stability of these solutions and disclose the effect of the dilaton and Born-Infeld fields on the thermal stability in the canonical ensemble.