  We investigate the temperature region in which a Tomonaga-Luttinger liquid
(TLL) description of the charge sector of the one-dimensional Hubbard model is
valid. By using the thermodynamic Bethe ansatz method, electron number is
calculated at finite temperatures and fixed chemical potential. We observe
maximum electron number as a function of temperature close to the chemical
potential of the upper critical value that corresponds to half filling. As the
chemical potential approaches the upper critical value from below, the
temperature $(T_{\rm M})$ at which the electron number shows its maximum
asymptotically approaches a universal relation. We show that, below the energy
corresponding to $T_{\rm M}$, the charge excitation spectrum nearly obeys a
linear dispersion relation. The results demonstrate that $T_{\rm M}$ marks the
important temperature below which TLL is realized.
