If you have a betta in a one gallon tank, I think too little water/space will get to it before temp being a few degrees off.
If I remember correctly, bettas like upper mid 70's to lower 70's most. If you want your betta to live 10 years, you will need a heater. But you will also need a filter and more space. I would suggest a 2.5 gallon tank minimum for optimal betta care. Mine is in a 2.5 gallon bowfront with a sponge filter. Very simple and I can adjust flow to suit the bettas needs.
If you want a heater, get at least a 2.5. For the cost of everything being extra small, you will be paying as much as you would for a similar 5.5 gallon setup.