Vacuum expectation values of the energy-momentum tensor and the Casimir forces are evaluated for scalar and electromagnetic fields in the geometry of two curved boundaries on background of the Robertson-Walker spacetime with negative spatial curvature. Robin boundary conditions are imposed in the case of the scalar field and perfect conductor boundary conditions are assumed for the electromagnetic field. We use the conformal relation between the Robertson-Walker and Rindler spacetimes and the corresponding results for two parallel plates moving with uniform proper acceleration through the Fulling-Rindler vacuum. For the general scale factor the vacuum energy-momentum tensor is decomposed into the boundary free and boundary induced parts. The latter is non-diagonal. The Casimir forces are directed along the normals to the boundaries. For Dirichlet and Neumann scalars and for the electromagnetic field these forces are attractive for all separations.
