In my link on Testosterone and the Adrenals I cover a very interesting study that showed that testosterone limited the adrenals responsiveness to ACTH. Of course, ACTH is released by the pituitary in high stress situations and I gave numerous examples from the Peak Testosterone Forum of low testosterone men feeling dramatically better after going on HRT and having their testosterone levels dramatically boosted. This is probably from this “stress dampening” effect of testosterone, although more research would be nice. (Testosterone also profoundly and positively affects the male brain.)
However, one question that I did not answer is “what effect does testosterone (therapy) have on cortisol?” Does it raise or lower cortisol levels?
Unfortunately, the answer to that question is not as easy to find as we might like. There is a study that looked at giving men testosterone enanthate, which is very similar to the testosterone cypionate that most American men on HRT are used to and found that “No obvious changes were found in the levels of C.” 
This was a small study on healthy men but shows the consensus: testosterone therapy seems to have no major effect on cortisol. However, I think that testosterone likely does affect cortisol in some rather key ways that I will mention below.
Again, though, finding larger studies on more varied populations is few and far between. I think we can still come to a fairly definite conclusion based on input from the steroid community as well.
First of all, there are quite a few stories out there of steroid users greatly lowering their cortisol levels during a cycle. However, I doubt that this is normative for men on any kind of reasonable testosterone replacement for a simple reason: if testosterone decreased cortisol, we would be reading about it everywhere. For example, if testosterone really lowered cortisol, why did we not see at least some small effect in the study above. And, let’s face it: it would be huge news if testosterone therapy substantially lowered cortisol levels and there are enough men on HRT now that this would almost for sure be a well-known observation.
1. Testosterone-to-Cortisol Ratio. Stop and think of the significance of the discussion above: if you go on HRT, you are very likely going to significantly increase your T/C, or testosterone-to-cortisol, ratio. Why? A hypogonadal male that doubles or even triples his testosterone without much change to his cortisol levels has clearly doubled or tripled his T/C ratio
2. Cortisol Reactivity. Cortisol or stress reactivity has a very specific medical meaning and I am not referring to that here. However, what I am trying to point out that is men on HRT who have, in fact, actually improved their testosterone numbers will very likely put out much less cortisol than they normally would have due to the ACTH effect mentioned above.  Of course, this assumes normal adrenal function.
So the bottom line is that you are not likely to get a big drop in cortisol in the short term from testosterone replacement, although anything is possible of course as we are all individual and the studies deal with averages of course..
LONG TERM: What happens to cortisol as the months and years roll by for men on testosterone therapy? Anecdotally, I have come across stories of men who claim that their cortisol eventually significantly decreased due to the “shutdown” effect of HRT on the HPT (hypothalmic-pituitary-testes) axis. I have seen no proof but have no doubt that it does occur in some men. Of course, the wise thing is to measure your cortsiol before and after starting testosterone therapy and include this in your regular monitoring program.
CAUTION: Going on or off HRT is a complex decision based on many factors and must be managed between you and a good physician. See my link on Hormone Replacement Therapy for some starter information.
1) International Journal of Andrology, August 1988, 11(4):265-276, “Endocrine, seminal and peripheral effects of depot medroxyprogesterone acetate and testosterone enanthate in men”
2) Neuropsychopharmacology, 2005 October, 30(10):1906 1912, “Testosterone Suppression of CRH-stimulated Cortisol in Men”