  Gravitational waves (GWs) from the compact binary coalescences can be used as
standard sirens to explore the cosmic expansion history. In the next decades,
it is anticipated that we could obtain the multi-band GW standard siren data
(from nanohertz to a few hundred hertz), which are expected to play an
important role in cosmological parameter estimation. In this work, we give for
the first time the joint constraints on cosmological parameters using the
future multi-band GW standard siren observations. We simulate the multi-band GW
standard sirens based on the SKA-era pulsar timing array (PTA), the Taiji
observatory, and the Cosmic Explorer (CE) to perform cosmological analysis. In
the $\Lambda$CDM model, we find that the joint PTA+Taiji+CE data could provide
a tight constraint on the Hubble constant with a $0.5\%$ precision. Moreover,
PTA+Taiji+CE could break the cosmological parameter degeneracies generated by
CMB, especially in the dynamical dark energy models. When combining the
PTA+Taiji+CE data with the CMB data, the constraint precisions of $\Omega_{\rm
m}$ and $H_0$ are $1.0\%$ and $0.3\%$, meeting the standard of precision
cosmology. The joint CMB+PTA+Taiji+CE data give $\sigma(w)=0.028$ in the $w$CDM
model and $\sigma(w_0)=0.11$ and $\sigma(w_a)=0.32$ in the $w_0w_a$CDM model,
which are comparable with or close to the latest constraint results by
CMB+BAO+SN. In conclusion, it is worth expecting to use the future multi-band
GW observations to explore the nature of dark energy and measure the Hubble
constant.
