  We show that the $\sigma/f_0(500)$ state with finite-temperature $T$
corrections to its spectral properties included, plays an essential role for
the description of the scalar susceptibility $\chi_S$, signaling chiral
symmetry restoration. First, we use the $O(4)$ Linear Sigma Model as a testbed
to derive the connection between $\chi_S$ and the $\sigma$ propagator and to
check the validity and reliability of the approach where $\chi_S$ is saturated
by the $\sigma/f_0(500)$ inverse self-energy, which we calculate at finite $T$
to one loop. A more accurate phenomenological description is achieved by
considering the saturation approach as given by the thermal $f_0(500)$ state
generated in Unitarized Chiral Perturbation Theory. Such approach allows to
describe fairly well recent lattice data within the uncertainty range given by
the UChPT parameters. Finally, we compare the UChPT saturated description with
one based on the Hadron Resonance Gas, for which the hadron mass dependences
are extracted from recent theoretical analysis. Several fits to lattice data
are performed, which confirm the validity of the thermal $f_0(500)$ saturated
approach and hence the importance of that thermal state for chiral symmetry
restoration.
