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In this work, extending a previous study at zero temperature ($T=0$), weperform a systematic study of the modifications to the QCD vacuum energydensity $\epsilon_{vac}$ in the finite-temperature case, above the chiraltransition at $T_c$, caused by a nonzero value of the parameter $\theta$, usingtwo different effective Lagrangian models which implement the $U(1)$ axialanomaly of the fundamental theory and which are both well defined also above$T_c$. In particular, we derive (and critically compare) the expressions forthe topological susceptibility $\chi$ and for the second cumulant $c_4$starting from the $\theta$ dependence of $\epsilon_{vac}(\theta)$ in the twomodels.

Theta dependence of the vacuum energy density in chiral effective Lagrangian models at finite temperature, above $T_c$