  We consider a scalar field model with a $g \phi_4^4$ interaction and compute
the mass correction at next-to-leading order in a large-$N$ expansion to study
the summability of the perturbative series. It is already known that at zero
temperature this model has a singularity in the Borel plane (a "renormalon").
We find that a small increase in temperature adds two countable sets both with
an infinite number of renormalons. For one of the sets the position of the
poles is thermal independent and the residue is thermal dependent. In the other
one both the position of poles and the residues are thermal dependent. If we
consider the model at extremely high temperatures, however, one observes that
all the renormalons disappear and the model becomes Borel summable.
