  We calculate, within the single-loop or equivalently the Hartree-Fock
Approximation (HFA), the finite-temperature interacting compressibility for
three-dimensional (3D) Dirac materials and renormalized quasiparticle
velocities for 3D and two-dimensional (2D) Dirac materials. We find that in the
extrinsic (i.e., doped) system, the inverse compressibility (incompressibility)
and renormalized quasiparticle velocity at $k=0$ show nonmonotonic dependences
on temperature. At low temperatures the incompressibility initially decreases
to a shallow minimum with a $T^2 \ln T$ dependence. As the temperature
increases further, the incompressibility rises to a maximum and beyond that it
decreases with increasing temperature. On the other hand, the renormalized
quasiparticle velocity at $k=0$ for both 2D and 3D Dirac materials first
increases with $T^2$, rises to a maximum, and after reaching the maximum it
decreases with increasing temperature. We also find that within the HFA, the
leading-order temperature correction to the low-temperature renormalized
extrinsic Fermi velocity for both 2D and 3D doped Dirac materials is $\ln
(1/T)$.
