  In a very recent paper [1], we have proposed a novel $4$-dimensional
gravitational theory with two dynamical degrees of freedom, which serves as a
consistent realization of $D\to4$ Einstein-Gauss-Bonnet gravity with the
rescaled Gauss-Bonnet coupling constant $\tilde{\alpha}$. This has been made
possible by breaking a part of diffeomorphism invariance, and thus is
consistent with the Lovelock theorem. In the present paper, we study
cosmological implications of the theory in the presence of a perfect fluid and
clarify the similarities and differences between the results obtained from the
consistent $4$-dimensional theory and those from the previously considered,
naive (and inconsistent) $D\rightarrow 4$ limit. Studying the linear
perturbations, we explicitly show that the theory only has tensorial
gravitational degrees of freedom (besides the matter degree) and that for
$\tilde{\alpha}>0$ and $\dot{H}<0$, perturbations are free of any pathologies
so that we can implement the setup to construct early and/or late time
cosmological models. Interestingly, a $k^4$ term appears in the dispersion
relation of tensor modes which plays significant roles at small scales and
makes the theory different than not only general relativity but also many other
modified gravity theories as well as the naive (and inconsistent) $D\to 4$
limit. Taking into account the $k^4$ term, the observational constraint on the
propagation of gravitational waves yields the bound $\tilde{\alpha} \lesssim
(10\,{\rm meV})^{-2}$. This is the first bound on the only parameter (besides
the Newton's constant and the choice of a constraint that stems from a temporal
gauge fixing) in the consistent theory of $D\to 4$ Einstein-Gauss-Bonnet
gravity.
