Does semi-classical gravity obey the equivalence principle? Question
I was recently wondering about semi-classical gravity :
$$ G_{\mu \nu} = \frac{8 \pi G}{c^4} \langle \hat T_{\mu \nu} \rangle_\psi$$
Does this obey the equivalence principle? 
My intuition
Let's say I am in a lift and I want to measure the standard deviation in the acceleration. Then with Heisenberg's equation of motion I can do so ,however , the left hand side $ G_{\mu \nu}$ is a classical object and will predict there is none whereas using the Hamiltonian in quantum mechanics $\hat T_{00}$ will only have zero standard deviation if it is in a acceleration eigenstate. But this view would only work in quantum mechanics and I'm not sure how to argue it for fields. Does this kind of argument still hold in QFT?
 A: Semiclassical gravity does violate the equivalence principle.
For example, consider the  effect of QED in a gravitational field on photon propagation, discovered by Drummond & Hathrell:


*

*Drummond, I. T., & Hathrell, S. J. (1980). QED vacuum polarization in a background gravitational field and its effect on the velocity of photons. Physical Review D, 22(2), 343, doi:10.1103/PhysRevD.22.343.


The effect could be understood in qualitative terms as a vacuum polarization  from virtual electron–positron pairs $e^-e^+$. These pairs would experience tidal effects from anisotropic curvature tensor with a characteristic length scale, the Compton wavelength of the electron. As a result electromagnetic field would now gain coupling to the curvature tensor. Therefore a photon propagating in a curved spacetime would  have a velocity dependent on its direction and polarization and differing from the “normal” speed of light $c$. We now have gravitational birefringence and the possibility of superluminal photons (such FTL photons would not violate causality).
Of course, such effects in a real astrophysical situations would be immeasurably tiny.
For example, in the gravitational field of a Schwarzschild black hole the characteristic relative difference between velocities of different polarizations
$$
\epsilon  \approx \frac{\alpha}{30 \pi } \frac{r_s ƛ^2 }{r^3} ,
$$
where $r_s$ is the Schwarzschild radius of a black hole, $ƛ$ is the (reduced) Compton wavelength of the electron, $\alpha$ is the fine structure constant. For a black hole of 1 solar mass this is about $10^{-36}$ near the horizon.
A: So it seems like your question and intuition are asking two separate questions. Your intuition is asking about the way measurements are made in the classical world and in the quantum world. Semi-classical theories are known to be incomplete and so some parts of the physical process won't actually stand up, ie matching standard deviations. 
That being said, we know GR is the most accurate field theory of gravity and most theories of gravity retain the equivalence principle in some form or another. In your question the equivalence principle is buried deep within the $G_{\mu\nu}$ and it's derivation. So with gravity, you will likely always use the equivalence principle it will just be disguised within the mathematics of the problem. But it is always there.
