As it happens, there is a teeny, tiny, itty-bitty, wee little grain of actual science behind this. Capsaicin, the active ingredient in cayenne pepper, is under investigation for its anti-inflammatory properties in helicobacter stomach ulcers.
http://www.ncbi.nlm.nih.gov/pubmed/17760719
So, the train of thought goes like this, I suppose: gum disease, inflammation, capsaicin, anti-inflammatory, rub cayenne on your inflamed gums.
However, the various dental associations websites that I briefly peruse do not say that anti-inflammatories are useful for periodontal disease. Cayenne might make your gums feel better, but it's not going to make your diseased gums stop receding.
Also, I note in passing that too much brushing is thought to contribute to periodontal disease. So brushing with cayenne pepper, if your gums are already at risk, doesn't sound like a good idea to me.