\begin{document}$\delta E_{\rm{shell}}$\end{document}, entropy \begin{document}$T \delta S_{\rm{shell}}$\end{document}, and free energy \begin{document}$\delta F_{\rm{shell}}$\end{document} is studied by employing the covariant density functional theory for closed-shell nuclei. Taking \begin{document}$^{144}$\end{document}Sm as an example, studies have shown that, unlike the widely-used exponential dependence \begin{document}$\exp(-E^*/E_d)$\end{document}, \begin{document}$\delta E_{\rm{shell}}$\end{document} exhibits a non-monotonous behavior, i.e., first decreasing 20% approaching a temperature of \begin{document}$0.8$\end{document} MeV, and then fading away exponentially. Shell corrections to both free energy \begin{document}$\delta F_{\rm{shell}}$\end{document} and entropy \begin{document}$T \delta S_{\rm{shell}}$\end{document} can be approximated well using the Bohr-Mottelson forms \begin{document}$\tau/\sinh(\tau)$\end{document} and \begin{document}$[\tau \coth(\tau)-1]/\sinh(\tau)$\end{document}, respectively, in which \begin{document}$\tau\propto T$\end{document}. Further studies on the shell corrections in other closed-shell nuclei, \begin{document}$^{100}$\end{document}Sn and \begin{document}$^{208}$\end{document}Pb, are conducted, and the same temperature dependencies are obtained."> Shell corrections with finite temperature covariant density functional theory -
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