The asymptotic expansion of the trace of the heat kernel Θ(t)=∑j=1∞​ exp(−tλj​) as t→0+ has been derived for a variety of domains, where {λj​} are the eigenvalues of the negative Laplace operator −Δ=−∑i=12​(∂xi∂​)2 in the (x1,x2)-plane. The dependence of Θ(t) on the connectivity of domains and the boundary conditions is analyzed. Particular attention is given for a general annular drum in R2 together with Robin boundary conditions, where the coefficients in the boundary conditions are positive smooth functions. Some applications of an ideal gas enclosed in the general annular drum are given