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The cosmological constant is the coefficient of a term in the Einstein tensor for which there are no a priori reasons to assume it to be zero, so that it could be regarded as a fundamental constant of nature. Mathematically it is indistinguishable however (at least in the Einstein equations) from a term in the stress-energy tensor originating from the vacuum expectation of a scalar field.

From what I understand, in quantum gravity, our graviton is the particle field associated to not directly the metric tensor, but rather its deviation from the background metric, which typically would be the Minkowski metric.

Now if we have a nonzero cosmological constant (as we appear to have), if we assume a scalar field as its origin, it seems reasonable to still consider Minkowski space to be the background, and the deviation of the metric tensor from its metric to be the graviton field.

However, if we have a cosmological term as a fundamental term in the Einstein tensor, it would seem more logical to take the appropriate (anti-)de Sitter space as the background, and treat the deviation of the metric tensor from its value as the graviton field.

Does the correct approach indeed depend on the origin of the cosmological term? Or is one of them always better, if so, which one, and why?

Thanks!

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A correct theory of quantum gravity must be background-independent, i.e. not depend on any particular "starting value" for the metric, the graviton field, or any other field. For more on background independence, see this question and its answers.

Perturbative formulations are always "naively background-dependent" in that they must expand around some background. In that case, background independence is the requirement that they must in principle yield the same results for every possible such background. So if the notion of your approaches even makes sense in a given speculative theory of quantum gravity, then they must be at least in principle equivalent. If one of them is in practice "better" (e.g. more computationally convenient) depends on the exact nature of the theory.

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