  The quantum critical behavior of the 2+1 dimensional Gross--Neveu model in
the vicinity of its zero temperature critical point is considered. The model is
known to be renormalisable in the large $N$ limit, which offers the possibility
to obtain expressions for various thermodynamic functions in closed form. We
have used the concept of finite--size scaling to extract information about the
leading temperature behavior of the free energy and the mass term, defined by
the fermionic condensate and determined the crossover lines in the coupling
($\g$) -- temperature ($T$) plane. These are given by $T\sim|\g-\g_c|$, where
$\g_c$ denotes the critical coupling at zero temperature. According to our
analysis no spontaneous symmetry breaking survives at finite temperature. We
have found that the leading temperature behavior of the fermionic condensate is
proportional to the temperature with the critical amplitude
$\frac{\sqrt{5}}3\pi$. The scaling function of the singular part of the free
energy is found to exhibit a maximum at $\frac{\ln2}{2\pi}$ corresponding to
one of the crossover lines. The critical amplitude of the singular part of the
free energy is given by the universal number
$\frac13[\frac1{2\pi}\zeta(3)-\mathrm{Cl}_2(\frac{\pi}3)]=-0.274543...$, where
$\zeta(z)$ and $\mathrm{Cl}_2(z)$ are the Riemann zeta and Clausen's functions,
respectively. Interpreted in terms the thermodynamic Casimir effect, this
result implies an attractive Casimir "force". This study is expected to be
useful in shedding light on a broader class of four fermionic models.
