they do, but most bettas can live just fine in 76F or so without compromising health. providing the water is clean etc.
most tropical fish are not fond of 80F...
infact many people keep shrimps and bettas together, when they are mostly cold water :3
I see :) it might be the preference of your own bettas too. My own has been in all kinds of temperatures from the low 70s to around 84F, he does slow down, but none of that is an indication of health. fish are cold blooded after all :) it only makes sense that their level of activity is controlled by the heat of the water. technically their lives spans too will be shortened in higher temps.
in all reality, water temperatures in the wild fluctuate. those who catch bettas in the wild can tell you the difference in water temperature between night and day :)
My bettas are happy and very active at 76-78... my thoughts are that unhappy bettas will not be active in any temperature. Until mine were moved into larger tanks (2 10g tanks, 1 7.9g) they spent a lot of time doing nothing, they'd mostly just find a spot and "park". Now that they're all in larger planted tanks they are much more active - I think they spend a lot of their time exploring (or doing a perimeter check) and only "park" to sleep.