The Cardy–Verlinde formula and topological AdS–Schwarzschild black holes

We consider the brane universe in the background of the topological AdS–Schwarzschild black holes. The induced geometry of the brane is that of a flat or an open radiation dominated FRW-universe. Just like the case of a closed radiation dominated FRW-universe, the temperature and entropy are simply expressed in terms of the Hubble parameter and its time derivative when the brane crosses the black hole horizon. We propose the modified Cardy–Verlinde formula which is valid for any values of the curvature parameter k in the Friedmann equations.