We study thermodynamic properties and phase structures of topological black holes in the Einstein theory with a Gauss-Bonnet term and a negative cosmological constant. The event horizon of these topological black holes can be an hypersurface with positive, zero or negative constant curvature. When the horizon is a zero curvature hypersurface, the thermodynamic properties of black holes are completely the same as those of black holes without the Gauss-Bonnet term, although the two black hole solutions are quite different. When the horizon is a negative constant curvature hypersurface, the thermodynamic properties of the Gauss-Bonnet black holes are qualitatively similar to those of black holes without the Gauss-Bonnet term. When the event horizon is a hypersurface with positive constant curvature, we find that the thermodynamic properties and phase structures of black holes drastically depend on the spacetime dimension d and the coefficient of the Gauss-Bonnet term: when d ≥ 6, the properties of black hole are also qualitatively similar to the case without the Gauss-Bonnet term, but when d = 5, a new phase of locally stable small black hole occurs under a critical value of the GaussBonnet coefficient, and beyond the critical value, the black holes are always thermodynamically stable. However, the locally stable small black hole is not globally preferred, instead a thermal anti-de Sitter space is globally preferred. We find that there is a minimal horizon radius, below which the HawkingPage phase transition will not occur since for these black holes the thermal anti de Sitter space is always globally preferred. *

Applying the first law of thermodynamics to the apparent horizon of a FriedmannRobertson-Walker universe and assuming the geometric entropy given by a quarter of the apparent horizon area, we derive the Friedmann equations describing the dynamics of the universe with any spatial curvature. Using entropy formulae for the static spherically symmetric black hole horizons in Gauss-Bonnet gravity and in more general Lovelock gravity, where the entropy is not proportional to the horizon area, we are also able to obtain the Friedmann equations in each gravity theory. We also discuss some physical implications of our results.