  We study the effective potential of a real scalar phi^4 theory as a function
of the temperature T within the simplest Phi-derivable approximation, namely
the Hartree approximation. We apply renormalization at a "high" temperature T*
where the theory is required to be in its symmetric phase and study how the
effective potential evolves as the temperature is lowered down to T=0. In
particular, we prove analytically that no second order phase transition can
occur in this particular approximation of the theory, in agreement with earlier
studies based on the numerical evaluation or the high temperature expansion of
the effective potential. This work is also an opportunity to illustrate certain
issues on the renormalization of Phi-derivable approximations at finite
temperature and non-vanishing field expectation value, and to introduce new
computational techniques which might also prove useful when dealing with higher
order approximations.
