We study the SU(N) symmetric model, which describes interaction of gravity with three field multiplets: first, the multiplet of pseudoscalar fields, which is now associated with the multi-component cosmic dark matter; second, the multiplet of vector fields, which represents the so-called color aether, now known as dynamic aether; third, the multiplet of Yang-Mills fields, which provides the SU(N) invariance of the model as a whole. It was previously known that the decay of the color aether in the early Universe could have given rise to emergence of an axionic singlet according to the Peccei-Quinn mechanism; we proposed an extended scheme, according to which the color aether activates an additional internal tool for generating not only a simple axionic singlet, but a whole SU(N) symmetric multiplet of pseudoscalar fields. Late-time evolution of the considered field configuration is analyzed in the framework of Bianchi-I cosmological model, and a hypothesis is proposed that the mentioned pseudoscalar multiplet can be associated with the multi-component cosmic dark matter.