  We prove nonlinear stability for a large class of solutions to the Einstein
equations with a positive cosmological constant and compact spatial topology in
arbitrary dimensions, where the spatial metric is Einstein with either positive
or negative Einstein constant. The proof uses the CMC Einstein flow and
stability follows by an energy argument. We prove in addition that the
development of non-CMC initial data close to the background contains a CMC
hypersurface, which in turn implies that stability holds for arbitrary
perturbations. Furthermore, we construct a one-parameter family of initial data
such that above a critical parameter value the corresponding development is
future and past incomplete.
