  Once the critical temperature of a cosmological boson gas is less than the
critical temperature, a Bose-Einstein Condensation process can always take
place during the cosmic history of the universe. Zero temperature condensed
dark matter can be described as a non-relativistic, Newtonian gravitational
condensate, whose density and pressure are related by a barotropic equation of
state, with barotropic index equal to one. In the present paper we analyze the
effects of the finite dark matter temperature on the properties of the
Bose-Einstein Condensed dark matter halos. We formulate the basic equations
describing the finite temperature condensate, representing a generalized
Gross-Pitaevskii equation that takes into account the presence of the thermal
cloud. The static condensate and thermal cloud in thermodynamic equilibrium is
analyzed in detail, by using the Hartree-Fock-Bogoliubov and Thomas-Fermi
approximations. The condensed dark matter and thermal cloud density and mass
profiles at finite temperatures are explicitly obtained. Our results show that
when the temperature of the condensate and of the thermal cloud are much
smaller than the critical Bose-Einstein transition temperature, the zero
temperature density and mass profiles give an excellent description of the dark
matter halos. However, finite temperature effects may play an important role in
the early stages of the cosmological evolution of the dark matter condensates.
