Varying gravitational constant $G(t)$ (VG) cosmology is studied in this paper, where the modified Friedmann equation and the modified energy conservation equation are given with respect to the constant-$G$ theory. Considering the extended Chaplygin gas (ECG) as background fluid (or thinking that ECG fluid is induced by the variation of $G$), the unified model of dark matter and dark energy is obtained in VG theory. The parameter spaces are investigated in the VG-ECG model by using the recent cosmic data. Constraint results show $\ensuremath{\beta}=\ensuremath{-}\frac{\stackrel{.}{G}}{HG}=\ensuremath{-}0.00{3}_{\ensuremath{-}0.020\ensuremath{-}0.055}^{+0.021+0.034}$ for the VG-GCG unified model and $\ensuremath{\beta}=\ensuremath{-}0.02{7}_{\ensuremath{-}0.032\ensuremath{-}0.066}^{+0.032+0.059}$ for the VG-MCG unified model. Equivalently, they correspond to the limits on the current variation of Newton's gravitational constant at 95.4% confidence level $|\frac{\stackrel{.}{G}}{G}{|}_{\text{today}}\ensuremath{\lesssim}4.1\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}12}\text{ }\text{ }{\mathrm{yr}}^{\ensuremath{-}1}$ and $|\frac{\stackrel{.}{G}}{G}{|}_{\text{today}}\ensuremath{\lesssim}6.6\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}12}\text{ }\text{ }{\mathrm{yr}}^{\ensuremath{-}1}$. And for $z\ensuremath{\le}3.5$, bounds on the variation of $\frac{\stackrel{.}{G}}{G}$ in the VG-ECG unified model are in accordance with the experiment explorations of varying $G$. In addition, in VG theory the used observational data point still cannot distinguish the VG-GCG and VG-MCG unified model from the most popular $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ cosmology. Furthermore, to see the effects of varying $G$ and physical properties for VG-ECG fluid, we discuss the evolutionary behaviors of cosmological quantities in VG theory, such as $\frac{\stackrel{.}{G}}{G}$, $\frac{\stackrel{..}{G}}{G}$ and equation of state $w$, etc. For $\ensuremath{\beta}l0$ a quintom scenario crossing over $w=\ensuremath{-}1$ can be realized in the VG-GCG model.
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