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 the boundary of 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 L2(D) satisfies a strong Hardy inequality with weight d^{-2}, (ii) the initial temperature distribution, and the specific heat of D are given by d^{-a} and d^{-b} respectively, where d is the distance to the boundary of D, and 1 < a < 2; 1 < b < 2.