  We develop the Hartree-Fock-Bogoliubov theory at finite temperature for Bose
gas trapped in the two dimensional optical lattices. The on-site energy is
considered low enough that the gas presents superfluid properties. We obtain
the condensate density as function of the temperature neglecting the anomalous
density in the thermodynamics equations. The condensate fraction provide two
critical temperature. Below the temperature $T_{C1}$ there is one condensate
fraction. Above two possible fractions merger up to the critical temperature
$T_{C2}$. Then the gas provides an first order transition at temperature above
$T_{C2}$ where the condensate fraction is null. We resume by a
finite-temperature phase diagram where can be identify three domains: the
normal fluid, the superfluid and the superfluid with two possible condensate
fractions.
