We present new measurements of the cosmic cold molecular gas evolution out to redshift 6 based on systematic mining of the Atacama Large Millimeter/submillimeter Array (ALMA) public archive in the COSMOS deep field (A3COSMOS). Our A3COSMOS data set contains ∼700 galaxies (0.3 ≲ z ≲ 6) with high-confidence ALMA detections in the (sub)millimeter continuum and multiwavelength spectral energy distributions. Multiple gas mass calibration methods are compared, and biases in band conversions (from observed ALMA wavelength to rest-frame Rayleigh-Jeans tail continuum) have been tested. Combining our A3COSMOS sample with ∼1000 CO-observed galaxies at 0 ≲ z ≲ 4 (75% at z < 0.1), we parameterize galaxies' molecular gas depletion time (τdepl) and molecular gas to stellar mass ratio (μmolgas) each as a function of the stellar mass (M∗), offset from the star-forming main sequence (ΔMS) and cosmic age (or redshift). Our proposed functional form provides a statistically better fit to current data (than functional forms in the literature) and implies a "downsizing" effect (i.e., more-massive galaxies evolve earlier than less-massive ones) and "mass quenching" (gas consumption slows down with cosmic time for massive galaxies but speeds up for low-mass ones). Adopting galaxy stellar mass functions and applying our function for gas mass calculation, we for the first time infer the cosmic cold molecular gas density evolution out to redshift 6 and find agreement with CO blind surveys as well as semianalytic modeling. These together provide a coherent picture of cold molecular gas, star formation rate, and stellar mass evolution in galaxies across cosmic time.