Nature of dark energy remains unknown. Especially, to constrain the time variability of the dark-energy, a new, standardisable candle that can reach more distant Universe has been awaited. Here we propose a new distance measure using fast radio bursts (FRBs), which are a new emerging population of $sim$ ms time scale radio bursts that can reach high-$z$ in quantity. We show an empirical positive correlation between the time-integrated luminosity (L$_{ u}$) and rest-frame intrinsic duration ($w_{rm int,rest}$) of FRBs. The L$_{ u}-w_{rm int,rest}$ correlation is with a weak strength but statistically very significant, i.e., Pearson coefficient is $sim$ 0.5 with p-value of $sim$0.038, despite the smallness of the current sample. This correlation can be used to measure intrinsic luminosity of FRBs from the observed $w_{rm int,rest}$. By comparing the luminosity with observed flux, we measure luminosity distances to FRBs, and thereby construct the Hubble diagram. This FRB cosmology with the L$_{ u}-w_{rm int,rest}$ relation has several advantages over SNe Ia, Gamma-Ray Burst (GRB), and well-known FRB dispersion measure (DM)-$z$ cosmology; (i) access to higher redshift Universe beyond the SNe Ia, (ii) high event rate that is $sim$ 3 order of magnitude more frequent than GRBs, and (iii) it is free from the uncertainty from intergalactic electron density models, i.e., we can remove the largest uncertainty in the well-debated DM-$z$ cosmology of FRB. Our simulation suggests that the L$_{ u}-w_{rm int,rest}$ relation provides us with useful constraints on the time variability of the dark energy when the next generation radio telescopes start to find FRBs in quantity.
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