dr liz, I grow French tarragon in the ground, not in a pot. But, I would guess that the roots are getting much too hot in that sized pot on a deck. You may need to find a way to cool the pot off (moving it under partial shade, watering twice a day - as long as drainage is excellent, etc.). Do you have a space where you can plant the French tarragon in the ground? It dies back to the roots each year, but comes back beautifully in the spring.

I second the above advice.