  We analyze in detail a behavior of the order parameter in the half-filled
extended Falicov-Kimball model for small Coulomb interactions (both onsite $U$
and intersite $V$). The parameter is defined as the difference of localized
electron concentrations in both sublattices of the Bethe lattice (in the limit
of large coordination number). Using two methods, namely, the dynamic mean
field theory and the Hartree-Fock approximation, we found the ranges of $U$ and
$V$ for which the anomalous temperature dependence of the order parameter,
characterized by the sharp reduction near $T \approx T_{\textrm{C}}/2$, occurs
($T_{\textrm{C}}$ is the temperature of the continuous order-disorder
transition). In order to quantitatively describe this anomaly, we defined a
function that measures the departure of the order parameter dependence from the
standard mean-field $S=1/2$ Ising-like curve. We determined the $U$-dependent
critical value $V_{\textrm{C}}$ of $V$ above which the anomaly disappears.
Indicators of the anomalous behavior of the parameter dependence can be also
observed in the temperature dependence of the specific heat.
