  It has been shown that time delay of strong gravitational lensing is not only
an effective cosmological probe to constrain the Hubble constant, the matter
density and the curvature of the universe but also useful for breaking the
degeneracy of the dark energy equation of state and thus provide
complementarity with other popular probes, such as type Ia supernovae, baryon
acoustic oscillations, and cosmic microwave background radiation.
Interestingly, compared to traditional strong lensing systems where quasars act
as sources, strongly lensed gravitational waves (GWs) from compact binary
coalescence and their electromagnetic (EM) counterparts systems have been
recently proposed to be more powerful for studying cosmology since GWs and
their EM counterparts are transients. Short durations of GWs and their EM
counterparts are very advantageous to time delay measurement and lens profile
modeling. Here, in the framework of Chevalier-Polarski-Linder (CPL)
parametrization, we investigate improvement of constraining power on dark
energy equation of state due to including time delay measurements of strong
lensed GW systems. It is suggested that, on the basis of the third generation
ground-based detector, e.g. Einstein Telescope, adding time delay of only 30
strong lensed GW systems to type Ia supernovae and cosmic microwave background
radiation can improve the dark energy figure of merit by a factor 2. For the
traditional standard siren method where the uncertainties of luminosity
distances of GWs are $\sim10\%$, a few $\times~10^4$ events are expected to
present similar constraints. In the precision cosmology era, this progress is
of great significance for studying the nature of dark energy.
