  Lattice QCD results for the coefficient $c_2$ appearing in the Taylor
expansion of the pressure show that this quantity raises with the temperature
towards the Stefan-Boltzmann limit. On the other hand, model approximations
predict that when a vector repulsion, parametrized by $G_V$, is present this
coefficient reaches a maximum just after $T_c$ and then deviates from the
lattice predictions. Recently, this discrepancy has been used as a guide to
constrain the (presently unknown) value of $G_V$ within the framework of
effective models at large-$N_c$ (LN). In the present investigation we show
that, due to finite $N_c$ effects, $c_2$ may also develop a maximum even when
$G_V=0$ since a vector repulsive term can be dynamically generated by exchange
type of radiative corrections. Here we apply the the Optimized Perturbation
Theory (OPT) method to the two flavor Polyakov--Nambu--Jona-Lasinio model (at
$G_V=0$) and compare the results with those furnished by lattice simulations an
by the LN approximation at $G_V=0$ and also at $G_V \ne 0$. The OPT numerical
results for $c_2$ are impressively accurate for $T \lesssim 1.2\, T_c$ but, as
expected, predict that this quantity develops a maximum at high-$T$. After
identifying the mathematical origin of this extremum we argue that such a
discrepant behavior may naturally arise within these effective quark models (at
$G_V=0$) whenever the first $1/N_c$ corrections are taken into account. We then
interpret this hypothesis as an indication that beyond the large-$N_c$ limit
the correct high temperature (perturbative) behavior of $c_2$ will be
faithfully described by effective models only if they also mimic the asymptotic
freedom phenomenon.
