  We study the RKKY interaction of magnetic impurities in the
$\alpha-\mathcal{T}_3$ model which hosts pseudospin-1 fermions with two
dispersive and one flat bands. By using the effective low-energy Hamiltonian we
calculate the RKKY coupling for impurities placed on the same or different
sublattices. We find that there are three types of interaction, which depend on
the model parameter defining the relative strength of hoppings between
sublattices, two of them can be reduced to graphene case while the third one is
new and is due to the presence of a flat zero-energy band. We derive general
analytical expressions for the RKKY interaction in terms of Mellin-Barnes type
integrals and analyze different limiting cases. The cases of finite chemical
potential and temperature, as well as asymptotic at large distances are
considered. We show that the interaction between impurities located at
different rim sites displays a very strong temperature dependence at small
doping being a direct consequence of the flat band. The subtleties of the
theorem for signs of the RKKY interaction at zero doping, as applied to the
$\mathcal{T}_3$ lattice, related to the existence of a dispersionless flat band
are discussed.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics