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The temperature dependence of the symmetry energy for isotopic chains ofeven-even Ni, Sn, and Pb nuclei is investigated in the framework of the localdensity approximation (LDA). The Skyrme energy density functional with twoSkyrme-class effective interactions, SkM* and SLy4, is used in thecalculations. The temperature-dependent proton and neutron densities arecalculated through the HFBTHO code that solves the nuclearSkyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformeddeformed harmonic-oscillator basis. In addition, two other densitydistributions of $^{208}$Pb, namely the Fermi-type density determined withinthe extended Thomas-Fermi (TF) method and symmetrized-Fermi local densityobtained within the rigorous density functional approach, are used. The kineticenergy densities are calculated either by the HFBTHO code or, for a comparison,by the extended TF method up to second order in temperature (with $T^{2}$term). Alternative ways to calculate the symmetry energy coefficient within theLDA are proposed. The results for the thermal evolution of the symmetry energycoefficient in the interval $T=0-4$ MeV show that its values decrease withtemperature. The temperature dependence of the neutron and protonroot-mean-square radii and corresponding neutron skin thickness is alsoinvestigated, showing that the effect of temperature leads mainly to asubstantial increase of the neutron radii and skins, especially in the moreneutron-rich nuclei, a feature that may have consequences on astrophysicalprocesses and neutron stars.

Temperature dependence of the symmetry energy and neutron skins in Ni, Sn, and Pb isotopic chains