  We study the cosmological solutions of the two-dimensional Brans-Dicke
equations considering a gas of $c=1$ strings in $S^{1}\times \mbox{\bf R}$ as
the source of the gravitational field. We also study the implications of the
$R$-duality invariance on the solutions. To this purpose we conjecture that, as
it happens for massless fields in finite boxes, the free energy of a gas of
massless string excitations is not given by the corresponding toroidal
compactification.
