Cosmological constant from the stress energy tensor or geometry? Sabine does make some interesting points. Can a cosmological constant come from the stress energy tensor? If so, I don't see how one is suppose to distinguish this as an all permeating field in the universe or just a constant fixed in the geometry side?
 A: If the constant really is a constant then you are correct that it is impossible to tell the difference between a constant on the left side that is a geometrical property and a constant on the right side that is a (constant) scalar field.
But why would  scalar field be constant unless it was a property of spacetime, in which case it has a natural interpretation as a geometrical property and is most obviously put on the left side? A scalar field on the right side would represent something like quintessence where the field could be a function of position and time. In that case the variation of the field could have observational consequences.
A: It really doesn't matter "which side of the equation" you put the cosmological constant on. The two sides suggest different points of view, which are good to know, but ultimately are completely equivalent (as you'd expect).

*

*If the cosmological constant term appears on the right-hand side, so you think of the cosmological constant as being part of the stress energy tensor, then it is natural to think of empty space as having a constant energy density determined by $\Lambda$. Quantum fluctuations are expected to contribute to the vacuum energy, leading to the cosmological constant problem.


*If the cosmological constant term appears on the left hand side, then you can think of Einstein's equations in vacuum (with no matter) as containing all the terms consistent with the symmetries of General Relativity, to second order in derivatives ("low energies" in an expansion of energy divided by the Planck mass). One way to phrase this is that "GR is the unique low energy theory of a massless spin-2 particle."
But, again, there's no difference in practice "on which side of the equation" the cosmological constant is located. The discussion is really meant as a provocative way to highlight different points of view on the physical interpretation of this term.
A: An interesting view on the cosmological constant$\Lambda$" describes professor Willie WY Wong in Traceless general relativity:
"... the constant Λ can be considered as a constant of integration (something that comes out of solving the equation) and not a fundamental constant of the laws of physics. This allows one to focus more on what the value of Λ is (something that can be determined by experiments) instead of why that value of Λ is what it is, and the equations of motion retain its scaling property...."
I think it supports John Rennie's answer.
