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According to \cite{Grande_2011} we can start by considering that the gravitational coupling and the vaccum energy density might evolve as a power series of some energy scale $\mu$ with rates given by  \begin{equation}  \label{variationrates}  \frac{d\rho_\Lambda(\mu)}{d \text{ ln } \mu} = \sum\limits_{k =0,1,2,...} A_{2k} \mu^{2k}, ~~ \frac{d}{d \text{ ln } \mu}\left( \frac{1}{G(\mu)} \right) = \sum\limits_{k =0,1,2,...} B_{2k} \mu^{2k} \mu^{2k}.  \end{equation}  We can regard this choice as purely phenomenological, and in this context a sensibile choice for the energy scale can be $H$. In this way we associate the running of the cosmological quantities to the typical energy scale of the gravitational field associated with the FLRW metric.  Let's first consider the evolution of $\rho_\Lambda$:  this type of expansion has been widely discussed in the literature and according to the results in \cite{Basilakos_2009}\cite{Babi__2002}\cite{Borges_2008} we can keep only the zeroth and second orderds to prevent deviations from the $\Lambda CDM$ model that are too big to be reconciled with current observations.  After integration we obtain the functional form:  \begin{equation}  \rho_\Lambda (H) = n_0 + n_2 H^2  \end{equation}  with the coefficient given by  \begin{equation}  n_0 = \rho_\Lambda^0 - \frac{3 \nu}{8 \pi} M_P^2 H_0^2, ~~~ n_2 = \frach{3 \nu}{8 \pi}M_P^2  \end{equation}  with  \begin{equation}  \nu = \frac{1}{6 \pi } \sum_i B_i \frac{M_i^2}{M_P^2}.  \end{equation}  Here $H_0$ and $\rho_\Lambda^2$ are respectively the the value of the Hubble rate and of the energy density of vaccum at the present time and the $M_i$ are the masses associated to every term in the expansion \ref{variationrates} in the underlying QFT derivation of the theory.  The parameter $nu$  \end{equation}