In theories of gravity with a positive cosmological constant, we consider
product solutions with flux, of the form (A)dS_p x S^q. Most solutions are
shown to be perturbatively unstable, including all uncharged dS_p x S^q
spacetimes. For dimensions greater than four, the stable class includes
universes whose entropy exceeds that of de Sitter space, in violation of the
conjectured "N-bound". Hence, if quantum gravity theories with
finite-dimensional Hilbert space exist, the specification of a positive
cosmological constant will not suffice to characterize the class of spacetimes
they describe.Comment: 25 pages; v2: references adde