Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?

Specifically, fix a definable forcing notion $\mathbb{P}$ - it is reasonable to ask how hard it would be to find a model where, say, $\Pi^1_3$ statements aren't changed by forcing with $\mathbb{P}$. For example, take $\mathbb{P}$ to be Cohen forcing. Then $L$ is certainly not such a model, but any forcing extension of $L$ by $\mathbb{P}$ is. On the other hand, there are certainly definable forcing notions such that the statement "Forcing with $\mathbb{P}$ doesn't change the truth value of $\Pi^1_3$ sentences" requires consistency strength beyond ZFC.

My question is when we can make "local absoluteness assumptions" without additional consistency strength. For a specific instance of this question:

Are there reasonable hypotheses on definable forcings $\mathbb{P}$ which imply that the statement $$\text{$PA(\mathbb{P})$ = "Projective sentences are absolute under forcing with $\mathbb{P}$"}$$ has no additional consistency strength?

I would ask for sufficient *and necessary* conditions, but I think that's demanding far too much.

*I've added the "descriptive set theory" tag because I think it may be relevant; feel free to delete if you think that's bonkers.*

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