  We calculate the finite temperature compressibility for two-dimensional
semiconductor systems, monolayer graphene, and bilayer graphene within the
Hartree-Fock approximation. We find that the calculated temperature dependent
compressibility including exchange energy is non-monotonic. In 2D systems at
low temperatures the inverse compressibility decreases first with increasing
temperature, but after reaching a minimum it increases as temperature is raised
further. At high enough temperatures the negative compressibility of low
density systems induced by the exchange energy becomes positive due to the
dominance of the finite temperature kinetic energy. The inverse compressibility
in monolayer graphene is always positive and its temperature dependence appears
to be reverse of the 2D semiconductor systems, i.e., it increases first with
temperature and then decreases at high temperatures. The inverse
compressibility of bilayer graphene shows the same non-monotonic behavior as
ordinary 2D systems, but at high temperatures it approaches a constant which is
smaller than the value of the non-interacting bilayer graphene. We find the
leading order temperature correction to the compressibility within Hartree-Fock
approximation to be $T^2 \ln T$ at low temperatures for all three systems.
