Usually, when we apply the non-relativistic limit ($c \rightarrow \infty$) to relativistic equations, the cosmological constant $\Lambda \sim \mathrm{L}^{-2}$ is simply offhandedly neglected by putting $\Lambda = 0$. I don't remember having saw other limits defined to $\Lambda$. However, it is theoretical possible to consider that this constant could depend on $c$ in such a way that a part of it could survive the non-relativistic limit. As basic examples, consider the following three "reparametrisations": \begin{equation} \Lambda = \begin{cases} \text{case $\mathcal{A}$ :} \quad \Lambda_0 \sim \mathrm{L}^{-2}. \\[12pt] \text{case $\mathcal{B}$ :} \quad \displaystyle{\frac{\Lambda_0}{c^2}}, \qquad \text{where $\Lambda_0 \sim \mathrm{T}^{-2}$.} \\[12pt] \text{case $\mathcal{C}$ :} \quad \displaystyle{\frac{\Lambda_0}{c^4}}, \qquad \text{where $\Lambda_0 \sim \mathrm{A}^2$ (squared acceleration).} \end{cases} \end{equation} The vacuum mass density is defined by the following expression: \begin{equation} \rho_{\text{vac}} = \frac{\Lambda c^2}{8 \pi G} = \begin{cases} \displaystyle{\frac{\Lambda_0 c^2}{8 \pi G}} \sim \frac{\mathrm{M}}{\mathrm{V}}. \qquad \text{There's no non-relativisitc limit in this case, if $\Lambda_0 \ne 0$.} \\[12pt] \displaystyle{\frac{\Lambda_0}{8 \pi G}} \sim \frac{\mathrm{M}}{\mathrm{V}}, \qquad \text{$\Lambda_0$ survives the non-relativistic limit.} \\[12pt] \displaystyle{\frac{\Lambda_0}{8 \pi G c^2}} \sim \frac{\mathrm{M}}{\mathrm{V}}, \qquad \text{$\Lambda_0$ is removed under the non-relativistic limit.} \end{cases} \end{equation} I guess case $\mathcal{C}$ should be the most natural formulation to properly apply the non-relativistic limit to Einstein's equation and recover Newton's gravitation theory. In this case, the parameter $\Lambda_0 \sim \mathrm{A}^2$ is associated to a purely relativistic phenomenon.
I now wonder if this kind of analysis has been discussed before and would like to know some references on it.
Can we define other limits as well (non-relativistic $c \rightarrow \infty$, non-gravitational $G \rightarrow 0$ or non-quantum $\hbar \rightarrow 0$ limits), by extracting more powers of $c$, and also extracting appropriate factors of $G$ and $\hbar$ from $\Lambda$? The resulting "classical" $\Lambda_0$ could then have some interesting interpretation.