  Upper bounds are obtained for the heat content of an open set D in a
geodesically complete Riemannian manifold M with Dirichlet boundary condition
on bd(D), and non-negative initial condition. We show that these upper bounds
are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting
in $L^2(D)$ satisfies a strong Hardy inequality with weight $r^2$, (ii) the
initial temperature distribution, and the specific heat of D are given by
$r^{-a}$ and $r^{-b}$ respectively, where $r$ is the distance to the boundary,
and 1<a<2, 1<b<2.
