Let's say we consider Einstein's Lagrangian from GR. In linearized gravity, we would expand the Ricci scalar to quadratic order in the perturbation parameter to find the propagator. My question is following :

Let's say we consider the Gauss Bonnet modification to GR, which is at the quadratic level. Hence, it must modify the propagator in some way. Can anyone show me how would it modify the GR propagator(with the calculations), and since Gauss Bonnet term is at higher derivative, my guess would be that the propagator would go as $\frac{1}{k^4}$ but would that not be acausal/non-local?

If you know a paper which has done these calculations in detail, it will be of great help.