The cosmological constant $\Lambda$ can be introduced into the gravitational action like this : \begin{equation} S = \frac{1}{2 \kappa} \int_{\Omega} (R - 2 \Lambda) \sqrt{-g} \; d^4 x + \text{matter terms}. \end{equation} The spacetime region $\Omega$ is arbitrary here. Now, what amaze me is that we can also write this : \begin{equation}\tag{1} -\: \frac{\Lambda}{8 \pi G} \int_{\Omega} \sqrt{-g} \; d^4 x = -\: \frac{\Lambda \, \mathcal{V}_4}{8 \pi G}, \end{equation} where $\mathcal{V}_4$ is the 4-volume of the spacetime region $\Omega$. So, the cosmological constant $\Lambda$ may be interpreted as the conjugate "variable" to $\mathcal{V}_4$, and as a Lagrange multiplier associated to a 4-volume constraint. We could suppose that since $\mathcal{V}_4$ should be very large and the action $S$ "reasonable", then $\Lambda$ should be small. My intuition tells me that there should be an equation like this one : \begin{equation}\tag{2} \Lambda \, \mathcal{V}_4 \sim \Lambda_{\text{max}} \mathcal{V}_{\text{min}}, \end{equation} where $\mathcal{V}_{\text{min}}$ is the smallest 4-volume that is physically meaningfull ; $\mathcal{V}_{\text{min}} \approx \ell_{\text{P}}^4$ ($\ell_{\text{P}}$ is the Planck length), and $\Lambda_{\text{max}} \approx \ell_{\text{P}}^{-2}$ is the "natural" value associated to the quantum vacuum. We then get \begin{equation}\tag{3} \Lambda \sim \frac{\ell_{\text{P}}^2}{\mathcal{V}_4}, \end{equation} which is thus very small. The relation $\Lambda \, \mathcal{V}_4 \approx \ell_{\text{P}}^2 \propto \hbar$ is also similar to an Heisenberg uncertainty relation ; $\Delta t \, \Delta E \ge \hbar$, which isn't surprising since the cosmological constant is introduced at the level of the action !
Can we make the previous idea more "rigorous" ? Does it make sense to interpret $\Lambda$ as a Lagrange multiplier associated to a constrained 4-volume introduced into the action ?
If the universe is spatially closed ($k = 1$) and also closed in time (especially if $\Lambda$ was negative), then the 4-volume of the whole universe would be finite.
Any opinion on this ?