  The physical Hamiltonian of a gravity-matter system depends on the choice of
time, with the vacuum naturally identified as its ground state. We study the
expanding universe with scalar field in the volume time gauge. We show that the
vacuum energy density computed from the resulting Hamiltonian is a non-linear
function of the cosmological constant and time. This result provides a new
perspective on the relation between time, the cosmological constant, and vacuum
energy.
