An equation of state in the limit of high densities

We take string theory in a box of volume $V$, and ask for the entropy $S(E,V)$. We let $E$ exceed the value $E_{bh}$ corresponding to the largest black hole that can fit in the box. Several approaches in the past have suggested the expression $S\sim \sqrt{EV/G}$. We recall these arguments, and in particular expand on an argument that uses dualities of string theory. We require that expression for $S(E,V)$ be invariant under the T and S dualities, and that it agree with the black hole entropy when $E\sim E_{bh}$. These criteria lead to the above expression for $S$. We note that this expression had been obtained also by a imposing a quite different requirement -- that the entropy within a cosmological horizon be of order the Bekenstein entropy for a black hole of size the cosmological horizon. We recall the earlier proposed model of a `dense gas of black holes' to model this entropy, and discuss its realization as a set of intersecting brane states. Finally we speculate that the cosmological evolution of such a phase may depart from the evolution expected from the classical Einstein equations, since the very large value of the entropy can lead to novel effects similar to the fuzzball dynamics found in black holes.