taller than wider tanks are not very good for bettas, they come from shallow waters and have wide territories so when picking a tank for them you should look for a wide tank. also something to keep in mind is it being large enough to heat, the smallest heatable size is 2.5gallons. basically, your tank is not suitable to keep a betta healthy.
I keep my bedroom around 80F to keep my aquariums around 78, so if you want to keep your house 80F you can do it that way, however other than that you'll just need to upgrade your tank (which you should do anyway). until you can, keep him on top of the fridge, it's a little warmer up there.