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)dSp×Sq. Most solutions are shown to be perturbatively unstable, including all uncharged dSp×Sq 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.