  We investigate the high temperature behavior of SU(5) in its minimal version.
We show that there exists a range of parameters of the Higgs potential for
which the symmetry remains broken at high temperature, thus avoiding the phase
transition that gives rise to the overproduction of monopoles . We also show
that in such scenario the thermal production of monopoles can be suppressed in
a wide range of parameters, keeping their number density below the cosmological
limits.
