  The compressibility, zero sound dispersion, and effective mass of a gas of
fermionic dipolar molecules is calculated at finite temperature for one-, two-,
and three-dimensional uniform systems, and in a multilayer
quasi-two-dimensional system. The compressibility is nonmonotonic in the
reduced temperature, $T/T_F$, exhibiting a maximum at finite temperature. This
effect might be visible in a quasi-low-dimensional experiment, providing a
clear signature of the onset of many-body quantum degeneracy effects. The
collective mode dispersion and effective mass show similar nontrivial
temperature and density dependence. In a quasi-low-dimensional system, the zero
sound mode may propagate at experimentally attainable temperatures.
