holonomy corrected Loop Quantum Cosmology (LQC), where the corresponding Friedmann equation depicts an ellipse on the plane $(H, \rho)$ [@singh; @singh1; @singh2; @singh3; @Haro1; @Haro2], being $H$ the Hubble parameter and $\rho$ the energy density. As shown in several papers, this simple background can be mimicked by modifying the Einstein-Hilbert action through the introduction of a term of the form $\sqrt{-g}f(\chi)$ where $g$ is the determinant of the metric, $f$ is a well-known function [@helling; @ds09; @Norbert; @langlois] and $\chi$ is a scalar, only depending on $H$ but not on its derivatives. The problem with this method is to actually find a scalar that for synchronous observers in the Friedmann-Lema[î]{}tre-Robertsont-Walker (FLRW) spacetime will only depend on the Hubble parameter.

From our viewpoint the simplest scalar is the extrinsic curvature [@ha17], which appears in a natural way when using the ADM formalism [@adm]. Disappointingly, this is not a covariant theory because the extrinsic curvature is not a true scalar in the sense that it depends on the slicing chosen and, thus, its use is only justified if there exists a preferred foliation of the spacetime. Following the spirit of [*Weyl’s principle*]{} (see [@rz] for a historical review), one could choose a 