Let $M$ be a smooth compact manifold of dimension $m$ with smooth, possibly empty, boundary $\partial M$. If $g$ is a Riemannian metric on $M$ and if $\nabla$ is an affine connection, let $D=D(g,\nabla)$ be the trace of the normalized Hessian; if $\partial M$ is empty, then we impose Dirichlet boundary conditions. The structures $(g,\nabla)$ arise naturally in the context of affine differential geometry and we give geometric conditions which ensure that $D$ is formally self-adjoint in this setting. We study the asymptotics of the heat equation trace; we have that $a_m(D)$ is an affine invariant. We use the asymptotics of the heat equation to study the affine geometry of affine hypersurfaces.