  The temperature dependence of the symmetry energy and symmetry free energy
coefficients of infinite nuclear matter and of finite nuclei is investigated.
For infinite matter, both these coefficients are found to have a weaker
dependence on temperature at densities close to saturation; at low but
homogeneous densities, the temperature dependence becomes stronger. For finite
systems, different definitions of symmetry energy coefficients are encountered
in the literature yielding different values. A resolution to this problem is
suggested from a global liquid-drop-inspired fit of the energies and free
energies of a host of nuclei covering the entire periodic table. The hot
nucleus is modeled in a subtracted finite-temperature-Thomas-Fermi framework,
with dynamical surface phonon coupling to nucleonic motion plugged in. Contrary
to infinite nuclear matter, a substantial change in the symmetry energy
coefficients is observed for finite nuclei with temperature.
