  We have conducted a thorough theoretical and numerical investigation of the
electronic susceptibility, polarizability, plasmons, their damping rates, as
well as the static screening in pseudospin-1 Dirac cone materials with a flat
band, or for a general $\alpha - \mathcal{T}_3$ model, at finite temperatures.
This includes calculating the polarization function, plasmon dispersions and
their damping rates at arbitrary temperatures and obtaining analytical
approximations the long wavelength limit, low and high temperatures. We
demonstrate that the integral transformation of the polarization function
cannot be used directly for a dice lattice revealing some fundamental
properties and important applicability limits of the flat band dispersions
model. At $k_B T \ll E_F$, the largest temperature-induced change of the
polarization function and plasmons comes from the mismatch between the chemical
potential and the Fermi energy. We have also obtained a series of closed-form
semi-analytical expressions for the static limit of the polarization function
of an arbitrary $\alpha - \mathcal{T}_3$ material at any temperature with exact
analytical formulas for the high, low and zero temperature limits which is of
tremendous importance for all types of transport and screening calculations for
the flat band Dirac materials.
