  Active galactic nucleus (AGN) feedback is expected to impact the amount of
cold gas in galaxies by driving strong galactic winds, by preventing external
gas inflows, or by changing the thermodynamical state of the gas. We use
molecular gas mass estimates based on dust absorption (H$\alpha$/H$\beta$) to
study gas content of large samples of type 2 AGN host galaxies in comparison
with inactive galaxies. Using sparse principal component and clustering
analysis, we analyze a suite of stellar and structural parameters of $\sim
27,100$ face-on, central galaxies at redshift $z = 0.02-0.15$ and with stellar
mass $M_\star \approx 10^{10}-2\times 10^{11}\,M_\odot$. We identify four
galaxy groups of similar mass and morphology (mass surface density, velocity
dispersion, concentration, and S\'{e}rsic index) that can be evolutionarily
linked through a lifecycle wherein gas content mediates their star formation
rate (SFR) and level of AGN activity. Galaxies first consume their gas mostly
through bursty star formation, then enter into a transition phase of
intermediate gas richness in which star formation and AGNs coexist, before
settling into retirement as gas-poor, quiescent systems with residual levels of
AGN activity (LINERs). Strongly accreting black holes (Seyferts) live in
gas-rich, star-forming hosts, but neither their gas reservoir nor their ability
to form stars seem to be impacted \emph{instantaneously} (timescales $\lesssim
0.5$\,Gyr) by AGN feedback. Our results are inconsistent with AGN feedback
models that predict that central, bulge-dominated, Seyfert-like AGNs in massive
galaxies have significantly lower molecular gas fractions compared to inactive
galaxies of similar mass, morphology, and SFR.
