Entropic gravity is like the "brazil nut effect" [0] [1]. The idea is that if you shake a glass full of different sized nuts, the large ones will rise to the top. From what I understand, this is because larger objects have more mass, moving slower when shaked, so as the larger (brazil nuts) don't move as much relative to the smaller ones (peanuts), and because of gravity, there's a cavity left under the brazil nut wh…
No, here "entropic" is as in the entropic force that returns a stretched rubber band to its unstretched condition, which (as it tends to be scrunched a bit) is at a higher entropy. https://en.wikipedia.org/wiki/Rubber_band_experiment "The stretching of the rubber band is an isobaric expansion (A → B) that increases the energy but reduces the entropy" [apologies for any reversed signs below, I think I caught them all]…
The paper cannot deal with fast-moving masses at all: it's not just the relativstic regime (where speeds are significant fractions of c) but rather the masses must move more slowly than the thermalization. This is hugely restrictive.
Finally, comparing themselves to the traditional approach of quantizing perturbations (e.g. turning classical (General Relativity) gravitational waves into lots of spin-2 gravitons) the authors write:
The gravitational interactions we observe at accessible
length scales could in principle emerge in many ways from
physics at the Planck scale ρ ∼ mPl/ℓ3 Pl ∼ 10104 J/cm3.
Perhaps the simplest is that gravitational perturbations
are quantized as gravitons, i.e., as another quantum field
theory like the gauge bosons of the other fundamental
forces in nature. This is a perfectly good effective quan-
tum field theory; nothing in principle forces us to aban-
don this picture until energies near the Planck scale.
They also say that while their starting point was being very different from the holographic picture: we find that the models have a range of free parameters,
and in some parameter regimes become indistinguishable
from standard virtual graviton exchange
Some of this will necessarily by driven by the need to be compatible with General Relativity in the weak field limit. They are not compatible with strong gravity in General Relativity at present.So while the idea is kinda interesting, I think they are putting the cart before the horse in asking what their model says about things like the interaction between gravitation and entanglement. That's simply unmeasurable by experiment right now whereas the very-well-understood relativistic precession of Mercury's perihelion is completely out of scope for this initial paper.