If there's no heater, then the outside air temperature will be the same as the water temperature. Generally, room temperature is far cooler than bettas need, especially at night. Do you know how much water is in the vase? If it's over two gallons, you should get a heater. If it's under two gallons, there's really no good way to keep a stable, suitable temperature without having an extremely stable room temperature of 78-80 (which most people don't). Also, I would be afraid of the oxygen content of the water (and air above the water) being too low. I would look into putting an airstone into the tank, as this would help circulate air in order to re-oxygenate the water.
Edit: Also, monthly water changes are not enough on nearly any tank unless it's vastly understocked. I'm not sure how much nitrate the plant can suck out of the water, but if the vase holds 2.5 gallons of water or less, I would do at least a 50% water change per week.