i don't think they do. they're very similar and related. though they're different.
cortisol (C) and testosterone (T) work hand in hand with one another. if your T level is high, C have to be much lower, they can't both be high at the same time. for example, if you're stressful, your C level will be high and your body automatically decreases T level to make it balance.
so you're on HRT, you won't suffer all the cortisol effects.