  We elucidate the properties of a gas of free closed bosonic strings in
thermal equilibrium. Our starting point is the intensive generating functional
of connected one-loop closed vacuum string graphs given by the Polyakov path
integral. Invariance of the path integral under modular transformations gives a
thermal duality invariant expression for the free energy of free closed strings
at finite temperature. The free bosonic string gas exhibits a self-dual
Kosterlitz-Thouless phase transition. The thermodynamic potentials of the gas
of free bosonic closed strings are shown to exhibit an infinite hierarchy of
thermal self-duality relations. Note Added (Sep 2005).
