MDF is practically purpose-made for this application. It is very dimensionally stable, one of its prime selling points. And while it's not as stiff or strong as ply it is as a rule more stable these days (because of the falling standard of plywood, while MDF is easier to make to a reasonable standard consistently).
As often mentioned MDF can bow under its own weight (a point regularly used with glee by its detractors). But that is when it is used flat as in a shelf, not edge-up. When it's oriented vertically it is much stronger and far less likely to sag, which is also true of any solid wood and ply incidentally.
3.6m (that's nearly 12 feet for Imperial users) is a very long span though, I hope the plan wasn't just to attach it at the ends and let the middle float. It shouldn't need much, just one or two points of attachment should do it.
An additional concern here is whether you can source MDF in a 3.6m length. If that's not the case the span won't be a single piece of material anyway, so a join of some kind will have to be devised and implemented. And of course any such joint will have to be factored in in how you attach the pelmet.
I've seen suggestions here that MDF is less likely to be warped (as long as it is kept dry) but I've also seen counter-examples:
Please note those don't give counter-examples to the quoted statement. Storage conditions that make for bent wood or manmade boards where gravity is a factor are completely separate to any tendency in a material to change dimensions or warp with changes in humidity.
Am I right in saying, if I buy some MDF from a hardware store, it is much more likely to be flat than ordering pineboard from a timberyard?
I wish for all our sakes that the answer to this was a simple yes, but unfortunately it's more complicated than that. It's not just the material qualities of the wood/board itself that is in play here, they way it's stored is hugely important.
Even stuff naturally un-inclined to bow can bow given half a chance by poor storage conditions. One great example of this is full sheets of ply only supported at or near both ends, even plywood 19mm and thicker can readily bow if stored this way, particularly in a stack.
And conversely weaker wood (e.g. long lengths of thin softwood) can remain just as flat as they came out of the kiln if stored well, i.e. fully supported along their length.