  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 $\sum m_\nu$ using the latest cosmological observations. In this
paper, some typical dark energy models, including $\Lambda$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 $\Lambda$CDM+$\sum m_\nu$ model, we obtain the upper limits of the
neutrino mass $\sum m_\nu < 0.123$ eV (DH), $\sum m_\nu < 0.156$ eV (NH), and
$\sum m_\nu < 0.185$ eV (IH) at the $95\%$ C.L., using the Planck+BAO+SNe data
combination. For the $w$CDM+$\sum m_\nu$ model and the CPL+$\sum m_\nu$ model,
larger upper limits of $\sum m_\nu$ are obtained compared to those of the
$\Lambda$CDM+$\sum m_\nu$ model. The most stringent constraint on the neutrino
mass, $\sum m_\nu<0.080$ eV (DH), is derived in the HDE+$\sum m_\nu$ 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.
