  The singular part of the finite-size free energy density $f_s$ of the O(n)
symmetric $\phi^4$ field theory in the large-n limit is calculated at finite
cutoff for confined geometries of linear size L with periodic boundary
conditions in 2 < d < 4 dimensions. We find that a sharp cutoff $\Lambda$
causes a non-universal leading size dependence $f_s \sim \Lambda^{d-2} L^{-2}$
near $T_c$ which dominates the universal scaling term $\sim L^{-d}$. This
implies a non-universal critical Casimir effect at $T_c$ and a leading
non-scaling term $\sim L^{-2}$ of the finite-size specific heat above $T_c$.
