  We analyze the leading terms of the spectral action for a model of
noncommutative geometry, which is a product of $4$-dimensional Riemannian
manifold with a two-point space exploring the previously neglected case when
the metrics over each sheet are different. Assuming the
Friedmann-Lema\^itre-Robertson-Walker type of the metric for both sheets we
obtain the action, which in addition to the the usual cosmological constant
terms and the Einstein-Hilbert term involves a nonlinear interaction term. We
study qualitative picture of potential consequences of such term in the basic
cosmological models.
