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Considering the mass splittings of three active neutrinos, we investigate how the nature of dark energy affects the cosmological constraints on the total neutrino mass $\sum m_ν$ using the latest cosmological observations. In this paper, some typical dark energy models, including $Λ$CDM, $w$CDM, CPL, and HDE models, are discussed. In the analysis, we also consider the effects from the neutrino mass hierarchies, i.e., the degenerate hierarchy (DH), the normal hierarchy (NH), and the inverted hierarchy (IH). We employ the current cosmological observations to do the analysis, including the Planck 2018 temperature and polarization power spectra, the baryon acoustic oscillations (BAO), the type Ia supernovae (SNe), and the Hubble constant $H_0$ measurement. In the $Λ$CDM+$\sum m_ν$ model, we obtain the upper limits of the neutrino mass $\sum m_ν< 0.123$ eV (DH), $\sum m_ν< 0.156$ eV (NH), and $\sum m_ν< 0.185$ eV (IH) at the $95\%$ C.L., using the Planck+BAO+SNe data combination. For the $w$CDM+$\sum m_ν$ model and the CPL+$\sum m_ν$ model, larger upper limits of $\sum m_ν$ are obtained compared to those of the $Λ$CDM+$\sum m_ν$ model. The most stringent constraint on the neutrino mass, $\sum m_ν<0.080$ eV (DH), is derived in the HDE+$\sum m_ν$ model. In addition, we find that the inclusion of the local measurement of the Hubble constant in the data combination leads to tighter constraints on the total neutrino mass in all these dark energy models.