A running cosmological constant $\Lambda(t)$ can always be included in the perfect fluid source tensor. By the transformation properties of tensors Einstein's field equation is still independent of the coordinates and therefore general covariant.

In "Dynamical system approach to running $\Lambda$ cosmological models" from Stachowski and Szydlowski the cosmological constants $\Lambda(t)=\Lambda(H(t))$ or $\Lambda(t)=\Lambda(a(t))$, which depends on the Hubble parameter $H(t)$ or scale factor $a(t)$, are labeled as non-covariant parametrizations. How could that be?


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