  We study some exact solutions in a $D(\ge4)$-dimensional Einstein-Born-Infeld
theory with a cosmological constant. These solutions are asymptotically de
Sitter or anti-de Sitter, depending on the sign of the cosmological constant.
Black hole horizon and cosmological horizon in these spacetimes can be a
positive, zero or negative constant curvature hypersurface. We discuss the
thermodynamics associated with black hole horizon and cosmological horizon. In
particular we find that for the Born-Infeld black holes with Ricci flat or
hyperbolic horizon in AdS space, they are always thermodynamically stable, and
that for the case with a positive constant curvature, there is a critical value
for the Born-Infeld parameter, above which the black hole is also always
thermodynamically stable, and below which a unstable black hole phase appears.
In addition, we show that although the Born-Infeld electrodynamics is
non-linear, both black hole horizon entropy and cosmological horizon entropy
can be expressed in terms of the Cardy-Verlinde formula. We also find a
factorized solution in the Einstein-Born-Infeld theory, which is a direct
product of two constant curvature spaces: one is a two-dimensional de Sitter or
anti-de Sitter space, the other is a ($D-2$)-dimensional positive, zero or
negative constant curvature space.
