  Recently a restriction ("quantum inequality-type relation") on the
(renormalized) energy density measured by a static observer in a "globally
static" (ultrastatic) spacetime has been formulated by Pfenning and Ford for
the minimally coupled scalar field, in the extension of quantum inequality-type
relation on flat spacetime of Ford and Roman. They found negative lower bounds
for the line integrals of energy density multiplied by a sampling (weighting)
function, and explicitly evaluate them for some specific spacetimes. In this
paper, we study the lower bound on spacetimes whose spacelike hypersurfaces are
compact and without boundary. In the short "sampling time" limit, the bound has
asymptotic expansion. Although the expansion can not be represented by locally
invariant quantities in general due to the nonlocal nature of the integral, we
explicitly evaluate the dominant terms in the limit in terms of the invariant
quantities. We also make an estimate for the bound in the long sampling time
limit.
