Perfect fluid tori with uniform distribution of the specific angular momentum orbiting the Kerr-de Sitter black holes or naked singularities are studied. Closed equipotential surfaces corresponding to stationary toroidal discs are allowed only in the spacetimes admitting stable circular geodesics. The last closed surface crosses itself in the cusp(s) enabling outflow(s) of matter from the torus due to the violation of hydrostatic equilibrium. The repulsive cosmological constant implies the existence ofthe outer cusp (with a stabilizing effect on the tori because of excretion, i.e., outflow of matter from the torus into the outer space) and the strong collimation of open equipotential surfaces along the rotational axis. Both the effects take place nearby so-called static radius where the gravitational attraction is just balanced by the cosmic repulsion.