Juuuuust to make sure, you aren't mistaking an agastache rupestris for a basil?
If not, heat like yours, I'd think it would do funny things to basil. :D plants aren't meant to grow in a blast furnace. (just kidding) so maybe???

Nope, it's basil. It started out fine, and it grows to over a foot every week after being cut, but the smell and flavor is off a tad.

Might it be thai basil? Or maybe a few seeds of that mixed in with regular basil?

Here is a link that might be useful: Thai Basil

There is a licorice basil (different from Thai basil). I bought seeds for it for next year when I went on a basil seed buying kick (ended up with 9 basil varieties)Help me, LOL. Do a search on it.

Basil seems to develop an off flavor if it's fertilized with too much nitrogen or too little phosphorus, potassium, and trace minerals. You could try digging some bone meal and greensand into the soil.