Unbounded entropy in spacetimes with positive cosmological constant

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.