I discuss a string-inspired cosmological model, in which primordial gravitational waves induce condensates of gravitational-anomaly terms, which are in turn responsible for inflation, and the generation of a (massless) gravitational axion background that breaks spontaneously Lorentz (and CPT) symmetry and remains undiluted at the end of inflation. The vacuum energy density of the system is of the type characterising ``a running vacuum model'', with terms proportional to the second and fourth power of the Hubble parameter, to leading order. As the Universe enters the radiation era, gravitational anomalies are cancelled by the generation of chiral fermionic matter at the end of inflation, as required for consistency of the matter quantum field theory. In that era, the undiluted gravitational-axion background induces tree-level leptogenesis in models with heavy right-handed neutrinos, which can then lead to baryogenesis (and thus matter-antimatter asymmetry), via sphaleron transitions in the standard model sector. At later eras of the Universe, chiral or QCD-type anomalous interactions of gauge potentials with the gravitational axion, which remain uncompensated after the cancellation of the gravitational anomalies by chiral matter, can generate - via non-perturbative (instanton) effects - a mass term for the axion, which can thus become a source for dark matter.