  The temperature dependence of the sigma meson and pion masses is studied in
the framework of the O(N) model. The Cornwall-Jackiw-Tomboulis formalism is
applied to derive gap equations for the masses in the Hartree and large-N
approximations. Renormalization of the gap equations is carried out within the
cut-off and counter-term renormalization schemes. A consistent renormalization
of the gap equations within the cut-off scheme is found to be possible only in
the large-N approximation and for a finite value of the cut-off. On the other
hand, the counter-term scheme allows for a consistent renormalization of both
the large-N and Hartree approximations. In these approximations, the meson
masses at a given nonzero temperature depend in general on the choice of the
cut-off or renormalization scale. As an application, we also discuss the
in-medium on-shell decay widths for sigma mesons and pions at rest.
