Coin raised a cluster of interesting issues for discussion, which need a separate thread.
(If we addressed them in the QG links thread it would interfere with its bibliography role.)
So I'm copying his post here:

Coin, as you indicate by mentioning some of it, there is a lot of research where people try modifying special relativity. DSR is a good keyword search term. Also searching for work by either of these two authors, Laurent F and Jerzy K-G, will bring up a lot on this. Just as an example here is a recent paper they co-authored

http://arxiv.org/abs/0710.2886 kappa-Minkowski space, scalar field, and the issue of Lorentz invariance
Laurent Freidel, Jerzy Kowalski-Glikman
Based on the talk given at the conference "From Quantum to Emergent Gravity", SISSA, June, 2007; to appear in the proceedings
(Submitted on 15 Oct 2007)

"We describe kappa-Minkowski space and its relation to group theory. The group theoretical picture makes it possible to analyze the symmetries of this space. As an application of this analysis we analyze in detail free field theory on kappa-Minkowski space and the Noether charges associated with deformed spacetime symmetries."

That paper would have references going back to earlier investigations of modified symmetry. I don't recommend reading the paper, but offer it as an example giving a window on current work.

The thread you mentioned in your post is one where John Baez joined in a lot, IIRC.
A student of his, Derek Wise, just did his PhD thesis in this area. Derek may also have posted in that thread. It might be something to look at.

Again as your post suggests, it is a complicated multifaceted business. I can't answer adequately, can only nibble away at it. This thread will not succeed unless several other people with expertise help out.

I will reply piecemeal some:

they don't say that they say deSitter SPACE is given by that coset space.

the deSitter GROUP is SO(4,1) and it contains "L" as a subgroup. whenever you have a subgroup you can take the quotient by it----unless the subgroup is normal (tech-term) the quotient won't be a group but it will inherit some useful structure. sometimes a quotient of Lie groups is called a homogeneous space. the original big group, in this case So(4,1) has a natural action on it. That's enough for me to say on this.

Oh, deSitter space (not the group, the space that the group is symmetries of) is like a hyperboloid of revolution.
(an hour glass? the waist of a cokebottle? an oldfashioned lady's corset? nothing seems quite as right as saying a hyperboloid of revolution. but the slices are 3-spheres instead of the circles that a usual hyperboloid gets sliced into.
the good Hurkyl who sometimes gets me out of muddles might show up----I think he did in that thread you linked to, or one like it.

Yes

Why does modified symmetry keep coming up in Quantum Gravity?

This is the main question that I think you are driving at. What all these modifications have in common is a relation to QG! And WHY does QG research seem to spawn all these ideas for modified symmetry?

One reason is MacDowell-Mansouri gravity (I think Laurent Freidel's involvement goes back to a Jan 2005 paper of his on Mac-Mans gravity). this gives a strong motivation for looking at SO(4,1).
My favorite paper about this is a 2006 paper by Derek Wise called something like MacDowell-Mansouri gravity and Cartan geometry
It is the paper with the picture of a hamster inside a plastic ball.
You know Baez has an exceptional flair for exposition and Derek Wise may also have some native talent, or developed some as Baez' student. I think it is brilliant expository writing. Mac-Mans is a FORMALISM for presenting GR in a different way that works elegantly and it has never been understood why it is so neat.

Another reason that modified symmetry of a great variety of sorts keeps coming up in QG is what I think is a kind of clunky mundane reason-----the persistent presence of the Planck length. You could say that the MISSION of QG is to resolve the bigbang and blackhole singularities and form ideas of what really happens down there where GR blows up-----those very small scale very high energy regimes.
So if there is going to be some new physics or some quantum corrections to GR they have to take effect at a certain SCALE.

But then this scale should look the same to all observers, right? At least that is what we are used to. It is intuitive that if new phenomena start happening at some scale then that scale should look the same, and should be invariant under the group.

OK well it turns out that you can modify the Lorentz group so that it keeps the speed c invariant for all observers and ALSO keeps a certain length invariant. That is what DSR is mostly about. You can see that it is motivated by QG.

BTW keeping c invariant doesnt mean you wouldnt ever see dispersion. c could be defined as the speed of light in the lowenergy limit, and the absolute top speed that light can go. But very high energy TeV gammaray could somehow interact with spacetime geometry itself and get slowed down somehow. Just slightly. Sounds crazy doesn't it? The group would allow c (the normal low energy limit for lightspeed) to be invariant and ALSO would allow for some significant length scale to be invariant.
For some years this perceived need to have a length scale (as well as a speed) be the same for all observers has been buzzing around annoying the hell out of QG people and goading them into trying desperate measures like DSR and kappaMinkowski etc.
And just this year in August there was the reported result from MAGIC gammaray telescope that they maybe maybe have observed a very tiny slowing down of TeV gammaray. Which tantalized people and infuriated them also, because the evidence was so iffy that they couldn't confidently believe in it. So that is still up in the air.

(and now Pereira Aldrovandi, the paper you linked, have retroactively PREDICTED the slight slowing down that MAGIC thought they saw, but weren't quite sure. so it is now even more up in the air)

Hope this helps some. Still a lot more to say about modified symmetry, extensions of relativity, as they relate to QG.

In case anyone is interested, a brief review of DSR by Jerzy K-G is here http://arxiv.org/abs/hep-th/0612280 Doubly Special Relativity at the age of six
Jerzy Kowalski-Glikman
To appear in the Proceedings of 22nd Max Born Symposium
(Submitted on 27 Dec 2006)

"The current status of Doubly Special Relativity research program is shortly presented.
I dedicate this paper to my teacher and friend Professor Jerzy Lukierski on occasion of his seventieth birthday."

I don't want to load you with reading. I mentioned the above links just in case somebody wants to get a taste of the literature.
the only paper I would recommend, which for some reason I am enthusiastic about, is:

http://arxiv.org/abs/gr-qc/0611154 MacDowell-Mansouri gravity and Cartan geometry
Derek K. Wise
34 pages, 5 figures
(Submitted on 30 Nov 2006)

"The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous 'model spacetime', including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A 'Cartan connection' gives a prescription for parallel transport from one "tangent model spacetime" to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as 'rolling' the model spacetime along physical spacetime. I explain Cartan geometry, and 'Cartan gauge theory', in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the context of Cartan geometry."

My personal 2 cents, my take on the whole topic for what it's worth, is that Pereira Aldrovandi are trying to make a deSitter General Relativity that will explain and predict stuff like TeV gammaray delay (if it is real) and accelerated expansion (if it is real) and that is an important initiative which will lead to testabilityhowever the formal way Per-Aldro do it might be somewhat ad hoc and their effort might benefit from some contact with Derek Wise because he also has a way, using Cartan geometry, of making a deSitter General Relativity.

To me this look very different from DSR, which was a modification of SPECIAL not general. I am not currently interested by DSR. What interests me is this current two-fold attempt to develop a (predictive) deS-GR.