Mostly I have little interest in this topic, but the discussion came up in a recent thread about what is the axiomatic basis of GR? I threw out what I would consider a 'physicist view': the action or field equation plus rules for relating mathematical object to natural objects are all you need; nothing else need be assumed. It was pointed out that this is not much like axioms in the mathematical sense of fundamental postulates from wich (hopefully) something substantive can be derived.
I bumped into a couple recent papers on axiomatic basis of GR. I haven't looked at them in great detail, but they might be interesting to those concerned with 'foundational issues'. One thing that comes out is that there are no axioms resembling any common statement of equivalence principle, general covariance, or that objects follow geodesics under any particular circumstances (except that an inertial observer is defined one following a timelike geodesic).
So, of possible interest:
Their references for related points seem mostly available in arxiv.
[EDIT: Actually, there is an analog of general covariance, accomplishing what Einstein intended, rather than what he wrote in his GR papers. It is not a stated axiom but is contained in the process of changing the SR axioms into the GR axioms].
[EDIT2]: I see at the very end that the view is not so different from mine. The Einstein field equations are added at the very end as either a definition(of the stress energy tensor) or another axiom, your choice.