This note by Alan Guth says that
The false vacuum, however, cannot rapidly lower its energy density, so the energy density remains constant and the total energy increases. Since energy is conserved, the extra energy must be supplied by the agent that pulled on the piston.
Why is it that the false vacuum cannot lower its energy rapidly?
Do we know of a substance which behaves in this unusual fashion? In short, why is the energy density doesn't dilute with expansion?