  The method of covariant symbols of Pletnev and Banin is extended to
space-times with topology $\R^n\times S^1\times ... \times S^1$. By means of
this tool, we obtain explicit formulas for the diagonal matrix elements and the
trace of the heat kernel at finite temperature to fourth order in a strict
covariant derivative expansion. The role of the Polyakov loop is emphasized.
Chan's formula for the effective action to one loop is similarly extended. The
expressions obtained formally apply to a larger class of spaces, $h$-spaces,
with an arbitrary weight function $h(p)$ in the integration over the momentum
of the loop.
