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Within the framework of the $U(3)$ chiral perturbation theory, we revisit themasses, decay constants and the mixing parameters of the light pseudoscalarmesons $\pi, K, \eta$ and $\eta'$. The low energy constants up tonext-to-next-to-leading order are determined by including the light-quark massdependences of the various quantities from different lattice QCD simulationsand relevant phenomenological inputs. Then we study the finite-temperaturebehaviors of the masses of the light pseudoscalar mesons. The thermal behaviorsof the $\eta$-$\eta'$ mixing angles in singlet-octet and quark-flavor bases arealso explored. While the masses of the $\pi, K$, and $\eta$ are increased whenincreasing the temperatures, the mass of the $\eta'$ turns out to be slightlydecreased in the low-temperature region.

Updated study of the $\eta$-$\eta'$ mixing and the thermal properties of light pseudoscalar mesons at low temperatures