  We present a string-based picture of the cosmological evolution in which
(CP-violating) gravitational anomalies acting during the inflationary phase of
the universe cause the vacuum energy density to "run" with the effective Hubble
parameter squared, $H^2$, thanks to the axion field of the bosonic string
multiplet. This leads to baryogenesis through leptogenesis with massive
right-handed neutrinos. The generation of chiral matter after inflation helps
in cancelling the anomalies in the observable radiation- and matter- dominated
eras. The present era inherits the same "running vacuum" structure triggered
during the inflationary time by the axion field. The current dark energy is
thus predicted to be mildly dynamical, and dark matter should be made of
axions. Paraphrasing Carl Sagan: "we are all anomalously made from starstuff " .
