  According to the standard prescriptions, zero-temperature string theories can
be extended to finite temperature by compactifying their time directions on a
so-called "thermal circle" and implementing certain orbifold twists. However,
the existence of a topologically non-trivial thermal circle leaves open the
possibility that a gauge flux can pierce this circle --- i.e., that a
non-trivial Wilson line (or equivalently a non-zero chemical potential) might
be involved in the finite-temperature extension. In this paper, we concentrate
on the zero-temperature heterotic and Type I strings in ten dimensions, and
survey the possible Wilson lines which might be introduced in their
finite-temperature extensions. We find a rich structure of possible thermal
string theories, some of which even have non-traditional Hagedorn temperatures,
and we demonstrate that these new thermal string theories can be interpreted as
extrema of a continuous thermal free-energy "landscape". Our analysis also
uncovers a unique finite-temperature extension of the heterotic SO(32) and
$E_8\times E_8$ strings which involves a non-trivial Wilson line, but which ---
like the traditional finite-temperature extension without Wilson lines --- is
metastable in this thermal landscape.
