The recently suggested generalized unimodular gravity theory, which was originally put forward as a model of dark energy, can serve as a model of cosmological inflation driven by the effective perfect fluid—the dark purely gravitational sector of the theory. Its excitations are scalar gravitons that can generate, in the domain free from ghost and gradient instabilities, the red tilted primordial power spectrum of cosmic microwave background (CMB) perturbations matching with observations. The reconstruction of the parametric dependence of the action of the theory in the early inflationary universe is qualitatively sketched from the cosmological data and satisfies the naturalness criterion—the O(1) magnitude of all its parameters. In particular, the only phenomenological exponentially big parameter—the e-folding factor with N∼60, leads in this model to a specific expression for tensor-to-scalar ratio r ≃ 0.004, matching with the existing observational bounds on r and the CMB spectral index. The alternative possibilities of generating the cosmological acceleration and possibility of generalized renormalizeability of the theory are also discussed.