I noticed this more than once. Whenever I'm in the range from 24mm to somewhere between 35mm and 50mm, the lens starts moving if I'm fixing it vertically with its front element facing the ceiling or the floor, is this a problem with my own lens or it's a general problem with this lens?
A quick search shows that you're not alone with this problem for that lens. (I don't have that lens myself, so I can't comment from experience). Since it is an L-series lens, you might want to just take it to Canon for them to service it.
Alternatively, if you want a DIY solution, the simplest fix might just be a thick rubber band or a piece of gaffer tape. Here's a collection of different DIY methods people have attempted for fixing the problem.
I have two of these lenses and have never experienced this. I would expect them to be stable, as L series lenses are expected to perform well. They are not the same though. One works with all my autofocus systems and the other only works on my 1DsMIII. I only bring this up to point out that not every lens is the same.
In my case, yes. Very typical. My EF 24-105 f/4L IS USM developed zoom creep after about four or five years. It's probably one of my most-used lenses. I did attempt peeling back the rubber on the zoom ring and using the screws to adjust the tension, but the creep returned pretty quickly. While a big flat rubber band, like the lensband, can be used to add enough friction to stop the ring from turning on its own when the zoom is "pulled" out by gravity, I found it enough of an irritation in use, that I just live with the creep.
Most not-top-of-the-line zoom lenses have this problem, as far as i know.
This is a guess, but i am assuming lenses which have no external moving parts might minimize this issue...but then again, this might have to do with the fact that these kinds of lenses are usually nearer to the top of the line.
To counteract it, fixing the lens on place with tape might be a quick and dirty solution (if the lens is retrofocal, which yours seems to be, and nothing moves externally while focusing).