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