I'll try to answer that via a couple of concrete examples of theories that deliberately set the speed of gravity and light at different values for good reasons.

The short answer is that General Relativity (GR) predicts that the speed of light is equal to the speed of gravitational waves. In order to have a difference between the speed, one needs a different theory from GR.

Moreover, the equality of the speed is well-tested at solar system scales; there is good evidence that at least within a few billion light years from here the speeds are the same; and there is good evidence that for at least the age of the solar system (several billion years) the speeds have been the same. A physically realistic theory has to match these lines of evidence.

Around General Relativity we've built the Standard Cosmology, and in that we construct a preferred frame of reference -- the comoving frame, or the cosmological frame -- in which we calculate the evolution of the size of the universe under the influence of attractive and repulsive matter. The comoving frame is the one containing all observers who see the universe as spatially flat, homogeneous, isotropic, and have a view pastwards to a hotter denser universe.

Our understanding of the universe decreases when the universe is very hot and dense, and our view that far pastwards with current equipment is obscured. How the early universe "behind the veil" of the oldest visible surface becomes that particular surface raises some tricky questions about the behaviour of the matter and spacetime in the very early universe.

Additionally we do not now have a microscopic description of all the attractive matter in the standard cosmology, and recent evidence suggesting an acceleration of the metric expansion raises questions about whether our microscopic description of repulisve matter is correct. Moreover, our macroscopic dsecsriptions of these still need some "help" in the early universe in order to produce homogeneity, isotropy and spatial flatness given the visible attractive matter and also the invisible bits of the Standard Model of particle physics, in the form of another form of repulsive matter.

From time to time it becomes fashionable or interesting to explore modifications to General Relativity that would eliminate the need for one or both types of repulsive matter. This is a trendy moment.

In General Relativity there is a universal coupling to the metric tensor. Gravitational waves are values propagating in this tensor field, and it is common to talk about them as propagating at the "tensor speed of sound".

One popular approach to modifying GR is to introduce a square scalar component that multiplies with the metric tensor to produce the gravitational action. Such theories are called scalar-tensor theories of gravity.

In scalar-tensor gravity you can make the tensor speed of sound unequal to the speed of light in a variety of ways.

One approach is to make the scalar value location-dependent, with its evolution modifying the gravitational attraction between clumps of matter.
In this sort of approach, one relates the scalar and tensor values to each other in such a way that one would still talk about just the "tensor speed of sound", with the scalar driving changes in that value (typically making the tensor sound speed slower in the past).

The phys.org describes a paper that deals with a subset of this type approach, which I'll get to further below.

Another approach is to have two metrics that do not couple universally. For instance, one could retain the metric tensor from General Relativity which couples to massive matter (like electrons and neutrinos) and add a metric scalar-tensor which couples to massless matter (like light and gluons). This lets the tensor speed of gravitational waves produced by light (and other massless particles) be slower than light.

Here's a concrete example of this approach:

Afshordi & Magueijo [ https://arxiv.org/abs/1603.03312 - I'll call this A&M] are exploring a (scalar-tensor) bimetric theory where, in the hot, dense, early universe the two metrics are unequal; massive matter couples to one metric, and (after electroweak symmetry breaking) while massless matter (mainly meaning light, since gluons stay confined in hadrons) couples to the other. The effect is that the sources of the former metric can move very slowly compared to light and the metric that it sources; it is just as valid to say that light moves much faster than, for example, the lightest massive particles. The result is that in the early universe, light can carry momentum-energy from one region of the early universe to another more efficiently than it can in the present day universe. Additionally, light zips ahead of its own gravity (or equivalently sheds cerenkov-like (tensor) gravitational radiation) and so even at the highest energies and energy-densities, light can homogenize the temperature of the early universe at extremely long length scales without disrupting structure formation and in particular without significantly changing the structure of the Cosmic Microwave Background.

After this temperature-evening process solves the horizon problem there is a phase change that causes the massless-matter metric to "freeze" into a configuration that is exactly identical to the massive-matter metric.

This leads to a present-day universe that reproduces all the observational and experimental successes of General Relativity and an early universe that the authors claim solves the horizon problem without cosmic inflation.

(They refer to other work on similar bimetric models that suggest that one or more other problems that cosmic inflation was introduced to solve -- homogeneity, isotropy and flatness -- might also be solved by their model).

Now onto the single scalar-tensor metric approach:

The Lombriser & Lima paper [ https://arxiv.org/abs/1602.07670 -- I'll call this L&L ] that's the subject of the phys.org article deals with a subset of Horndeski gravity theories. Horndeski theory are a general family of scalar-tensor theories that become General Relativity when the scalar freezes. It is very general and the scalar can relate among other things to a f(R) gravity -- a function on the Ricci scalar R -- or a f(\hat{G}) gravity -- a "variable G" theory, where G is Newton's constant. L&L focus on Horndeski theories that seek to eliminate the need for a cosmological constant term (or similar dark energy source term in the cosmological frame).

(Aside: A&M somewhat distinguish themselves from Horndeski gravities starting at the top of their p. 2. They do not use the vocabulary of Horndeski theory or expand on their argument, and additionally the A&M paper itself is focused on cosmic inflation rather than dark energy. So the applicability of the L&L argument is not especially clear.)

L&L argue that generically in a Horndeski theory that is able to reproduce the metric expansion of space, the evolving scalar induces a damping of (tensor) gravitational waves different from that of General Relativity; in particular, in order to get rid of dark energy you have to slow down tensor waves in the earlier universe, and speed them up in the later universe. This lets galaxy clusters separate from each other faster in the later universe than they do in the earlier universe. But this approach also produces observables in electromagnetically bright events that also produce detectable gravitational waves (e.g. inspiralling neutron stars that explode into a GRB or type II supernova and possibly shedding gravitational waves from fallback accretion onto the remnant(s)). Moreover, such events for a wide range of theories are within the threshold of forthcoming gravitational wave observatories.

Additionally in these theories, as light would zip along faster than the tensor speed of sound, they would shed Cerenkov-like (tensor) gravitational waves, which may have [a] their own observables but [b] reduce the speed of light relative to the speed of gravity when the path the light takes through spacetime is long. The authors argue that studies of objects with known spectra at different high redshifts should directly test this aspect of dark-energy-free Horndeski gravities.

Finally, the L&L paper doesn't look too deeply at the semiclassical picture, asking about whether other massless particles -- gluons, namely -- racing ahead of gravity should produce observables. Neither does A&M's paper. And both papers stay far away from questions of quantum gravity.