Let g(t) be a time-dependent family of Riemannian metrics on a manifold M w
ith a smooth boundary. Let phi be the initial temperature of M and let rho
be the specific heat of M. Impose Dirichlet or Neumann boundary conditions
and let beta(t) be the resulting total heat energy content of M. As t down
arrow 0, one can expand beta similar to Sigma(n) beta(n)t(n/2) in an asympt
otic series in half integer powers of the parameter t. We determine B-n for
n less than or equal to 4 in terms of geometric quantities; this extends p
revious results from the autonomous setting where the metric was independen
t of the parameter t to a dynamic setting where the metric is permitted to
be time dependent.