  Recently a new 4D Einstein-Gauss-Bonnet theory has been introduced
\textbf{[Phys. Rev. Lett. 124 (2020) 081301]} with a serious debate that it
does not possess a covariant equation of motion in $4D$. This feature, makes
impossible to consider non-symetric space-times in this model, such as
anisotropic cosmology. In this note, we will present a new proposal to make
this happen, by introducing a Lagrange multiplier to the action which
eliminates the higher dimensional term from the equation of motion. The theory
has then a covariant $4D$ equation of motion which is useful to study the less
symmetric metrics. On top of FRW universe, the constraint theory is equivalent
to the original $4D$ Einstein-Gauss-Bonnet gravity. We will then consider the
anisotropic cosmology of the model and compare the theory with observational
data. We will see that the theory becomes non-conservative and the matter
density abundance falls more rapidly at larger redshifts compared to the
conservative matter sources.
