  We investigate the impacts of the gravitational-wave (GW) standard siren
observation of the Einstein Telescope (ET) on constraining the total neutrino
mass. We simulate 1000 GW events that would be observed by the ET in its
10-year observation by taking the standard $\Lambda$CDM cosmology as a fiducial
model. We combine the simulated GW data with other cosmological observations
including cosmic microwave background (CMB), baryon acoustic oscillations
(BAO), and type Ia supernovae (SN). We consider three mass hierarchy cases for
the neutrino mass, i.e., normal hierarchy (NH), inverted hierarchy (IH), and
degenerate hierarchy (DH). Using Planck+BAO+SN, we obtain $\sum m_\nu<0.175$ eV
for the NH case, $\sum m_\nu<0.200$ eV for the IH case, and $\sum m_\nu<0.136$
eV for the DH case. After considering the GW data, i.e., using
Planck+BAO+SN+GW, the constraint results become $\sum m_\nu<0.151$ eV for the
NH case, $\sum m_\nu<0.185$ eV for the IH case, and $\sum m_\nu<0.122$ eV for
the DH case. We find that the GW data can help reduce the upper limits of $\sum
m_\nu$ by 13.7%, 7.5%, and 10.3% for the NH, IH, and DH cases, respectively. In
addition, we find that the GW data can also help break the degeneracies between
$\sum m_{\nu}$ and other parameters. We show that the GW data of the ET could
greatly improve the constraint accuracies of cosmological parameters.
