The rationale for claiming that gravitational waves travel at the speed of light is that the "carrier" of the gravitational interaction is a massless graviton, like that of the photon for electromagnetic interactions, and so, too, should travel at maximum speed.
The speed of light, c, refers to electromagnetic permeabilities, which determine how elastic the medium is, and has absolutely nothing to do with neutral matter, as that which would be the source of gravitational waves.
Gravitational waves are assumed to be transverse waves, just like electromagnetic waves where the polarizations occur in the plane perpendicular to the direction of the propagation of the wave. Whereas the polarization of light refers to 3D space, the polarization of gravitational waves refer to 4D space. This is nonsense, since polarization is a spatial phenomenon, and does not involve the time axis. When this is realized, Eddington's claim that TL and LL waves travel at the "speed of thought" loses all meaning.
If "spacetime" is different than the vacuum in which electromagnetic waves propagate, then they share no common speed, c. If they are the same, electromagnetic waves do not propagate in vacuum, and all our physical foundation of electromagnetism must be shelved.
Gravitation is known to affect the propagation of all types of waves, even electromagnetic ones. Then why is this not reciprocal? i.e., electromagnetic waves should affect gravitational waves with the consequence that the LIGO instruments don't measure what they were intended to measure, i.e., the measurement of the displacement of mirrors in an interferometer using laser light.
Gravitational waves are said to be observed when two black holes merge. The merger of two black holes is not geodesic motion, and, therefore, lies outside the domains of general relativity. The waves are source free solutions of the wave equation that results when the Ricci tensor is linearized. It is known that there are no periodic solutions to the nonlinear Einstein field equations. So, the linearized solution does not refer to a weak field of some more general wave equation (e.g., solitary waves). If the merger of black holes did not occur so far away, the radiation emitted would be much stronger. What, then, would be the equation governing the propagation of gravitational radiation?
In traversing a material medium the GW displaces from their equilibrium position elements of the material transversely to its direction of propagation. Consider a room in which a chandelier is hanging from the wall. A seismic S-wave traverses the room, and the displacement of the chandelier from its equilibrium is intended to be measured by LIGO. However, the whole room, and not just the chandelier, feels the effect of the passage of the S-wave. The measurement of the displacement of the chandelier means nothing since everything in the room has been affected, including the electromagnetic wave that was sent out to measure the displacement.
The "fabric" of spacetime that supposedly supports GW requires a Poisson ratio of 1. The transverse shear velocity will thus coincide with the speed of light, while the longitudinal, pressure, wave velocity will vanish since the bulk modulus vanishes. In general, the longitudinal wave velocity will be greater than the transverse wave velocity since the force is in the direction of propagation in contrast to being in the plane normal to propagation. This conclusion rests upon the fact that GW are polarizable. But the polarizability in 4D is meaningless.
The analogy between the Cauchy stress tensor and strain in terms of Hooke's law where the spring constant is proportional to Young's modulus, and the Einstein field equations highlights the imperfections in the latter. What would be analogous to Young's modulus, Y, is Einstein's constant k=8pi G/c^4, but the latter must be multiplied by the fourth power of the frequency, f, viz, Y=kf^4. This would mean that the ratio of the Einstein tensor T, to the energy-stress tensor is frequency dependent. To salvage the constitutive relation relating stress to deformation, which is what Cauchy's law is all about, and relate it to Einstein's field equation requires a bending deformation to replace straightforward stretch, contraction, or shear [Tenev & Hostemeyer, "The Mechanics of Spacetime"] And even if it could be salvaged, it would have only physical meaning in the (3+1) Numerical relativity formulation in which time lapse is separated from spatial extent. That is, space time is foliated into space like surfaces like the peel of an orange where the sections of the peel are related by time lapse functions. This would be catastrophic to the interpretation of the strain, i.e. the Einstein tensor, linearized into a sourceless wave equation since the stress, i.e., the energy-stress tensor would vanish.
Even if a fabric of space time would exist, it must necessary explain why the speed of propagation of waves is the same as the vacuum of electromagnetism where the speed of light is related to the "electrical elasticity" of the medium, which has been known since the time of Maxwell (1865). It will, therefore, certainly not be acceptable to state that since the graviton is massless like the photon, and GW and EM waves are polarizable, they should have a common speed between them.