  Multi-band gravitational-wave (GW) standard siren observations are poised to
herald a new era in the study of cosmic evolution. These observations offer
higher signal-to-noise ratios and improved localizations compared to those
achieved with single-band GW detection, which are crucial for the cosmological
applications of dark sirens. In this work, we explore the role multi-band GW
synergetic observations will play in measuring cosmological parameters,
particularly in comparison with single GW observatory data. We used mock
multi-band dark siren data from third-generation GW detectors and the baseline
Decihertz Interferometer Gravitational-Wave Observatory to infer cosmological
parameters. Our analysis was conservative, involving only the 89 actual GW
events from the current Gravitational Wave Transient Catalogs in our data
simulation, facilitating a direct comparison with existing dark siren results.
Multi-band GW observations significantly improve sky localization accuracy by
two to three orders of magnitude over single-band observations, although their
impact on luminosity distance error remains limited. This results in a
substantial improvement in the constraints on matter density and the Hubble
constant, enhancing them by $75\%$ to $85\%$ and $65\%$ to $82\%$,
respectively. We conclude that the significant potential of multi-band GW
synergistic observations for detecting GW signals and resolving the Hubble
tension is highly promising and warrants anticipation.
