  We examine the leading order semiclassical gradient corrections to the
non-interacting kinetic energy density functional of a two dimensional Fermi
gas by applying the extended Thomas-Fermi theory at finite temperature. We find
a non-zero von Weizs\"acker-like gradient correction, which in the
high-temperature limit, goes over to the familiar functional form
$(\hbar^2/24m) (\nabla\rho)^2/\rho$. Our work provides a theoretical
justification for the inclusion of gradient corrections in applications of
density-functional theory to inhomogeneous two-dimensional Fermi systems at any
{\em finite} temperature.
