Mulch may be mulch, but I don't think of stones as mulch. I agree with Ken that the stones would radiate heat. Think of it this way. If you were to touch a stone sitting in the Texas sun, would it be hot to the touch? Hot enough on some days to fry an egg (or at least make the egg whites turn white)? Perhaps so. Now think about touching a chunk of pine bark mulch, straw, or leaf litter sitting in that same sunlight. Would it feel as hot to the touch as the stone did? Could you fry an egg on it? Probably not.
As for the original question: I recommend getting a plant stake, rod, or dowel and pushing it deep into the ground right beside the rose's main cane. Then tie the stake to the rose cane to help it stand straight and upright. I like to use zip-ties, but you can also use nylon twine or green plant tape.
