  We study heat equations $\partial_t u + Lu = 0$ on bounded Lipschitz domains
$\Omega$, where $L=-\operatorname{div}(A\nabla\;\cdot\;)$ is a second-order
uniformly elliptic operator with generalised Robin boundary conditions of the
form $\nu\cdot A\nabla u + Bu=0$, where $B\in\mathcal{L}(L^2(\partial\Omega))$
is a general operator. In contrast to large parts of the literature on
non-local Robin boundary conditions we also allow for operators $B$ that
destroy the positivity preserving property of the solution semigroup
$(e^{-tL})_{t\ge 0}$. Nevertheless, we obtain ultracontractivity of the
semigroup under quite mild assumptions on $B$. For a certain class of operators
$B$ we demonstrate that the semigroup is in fact eventually positive rather
than positivity preserving.
