  We study in random-phase approximation the newly discovered supersolid phase
of ${}^4$He and present in detail its finite temperature properties. ${}^4$He
is described within a hard-core quantum lattice gas model, with nearest and
next-nearest neighbour interactions taken into account. We rigorously calculate
all pair correlation functions in a cumulant decoupling scheme. Our results
support the importance of the vacancies in the supersolid phase. We show that
in a supersolid the net vacancy density remains constant as function of
temperature, contrary to the thermal activation theory. We also analyzed in
detail the thermodynamic properties of a supersolid, calculated the jump in the
specific heat which compares well to the recent experiments.
