  We analyze the $f_0(500)$ state generated as a pole of $\pi\pi$ scattering
within unitarized low-energy effective theories at finite temperature. First,
we report on the $\sigma$ self-energy in the LSM, which is used as a benchmark
in order to test the hypotheses carried out. The relation of that thermal pole
with the scalar susceptibility is studied within a scalar saturation approach,
which yields results complying with lattice data. The robustness and
predictability of this method are studied in terms of the low-energy constants
involved and the unitarization method. Our analysis highlights the importance
of this thermal state to describe the main qualitative features of the scalar
susceptibility around the chiral transition.
