  We consider the $O(N+1)/O(N)$ Non-Linear Sigma Model for large $N$ as an
effective theory for low-energy QCD at finite temperature $T$, in the chiral
limit. At $T=0$ this formulation provides a good description of scattering data
in the scalar channel and generates dynamically the $f_0(500)$ pole, the pole
position lying within experimental determinations. Previous $T=0$ results with
this model are updated using newer analysis of pion scattering data. We
calculate the pion scattering amplitude at finite $T$ and show that it
satisfies exactly thermal unitarity, which had been assumed but not formally
proven in previous works. We discuss the main differences with the $T=0$ result
and we show that one can define a proper renormalization scheme with $T=0$
counterterms such that the renormalized amplitude can be chosen to depend only
on a few parameters. Next, we analyze the behaviour of the $f_0(500)$ pole at
finite $T$, which is consistent with chiral symmetry restoration when the
scalar susceptibility is saturated by the $f_0(500)$ state, in a second-order
transition scenario and in accordance with lattice and theoretical analysis.
