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We present the temperature and density dependence of symmetry energy fornuclear matter at finite temperature based on the approach of thethermodynamics with Skyrme energy density functional. We first classify theSkyrme interactions into 7 groups according to the range of neutron and protoneffective mass in neutron matter limit(99.99 per cent neutron in the matter).We find that there is obvious correlation between the temperature dependence ofthe symmetry energy and the splitting of the neutron and proton effective mass.For some Skyrme interactions with $m^{*}_{n}>m^{*}_{p}$ and strong splitting ofthe neutron and proton effective mass in asymmetric nuclear matter, atransition of the temperature dependence of symmetry energy from decreasingwith temperature at low densities to increasing with temperature at highdensities appears. For other Skyrme interactions, we do not observe suchphenomenon. Our study show that the symmetry energy in hot asymmetric matternot only depends on symmetry potential part but also on the splitting of theneutron and proton effective mass to a certain extent.

Effect of splitting of the neutron and proton effective mass on nuclear symmetry energy at finite temperature