We examine the cosmological effects of the Hagedorn phase in models where the
observable universe is pictured as a D-brane. It is shown that, even in the
absence of a cosmological constant, winding modes cause a negative `pressure'
that can drive brane inflation of various types including both power law and
exponential. We also find regimes in which the cosmology is stable but
oscillating (a bouncing universe) with the Hagedorn phase softening the
singular behavior associated with the collapse.Comment: 44 Pages; JHEP latex; includes 1 postscript figur