Heat-kernel expansion at finite temperature

We show how the zero-temperature result for the heat-kernel asymptotic expansion can be generalized to the finite-temperature one. We observe that this general result depends on the interesting ratio $\frac{\sqrt{\tau }}{\beta }$, where $\tau$ is the regularization parameter and $\beta =\frac{1}{T}$, so that the zero-temperature $\mathrm{limit} \beta \to \infty$ corresponds to the "cutoff" $\mathrm{limit} \tau \to 0$. As an example, we discuss some aspects of the axial model at finite temperature.