There is a lot of emerging evidence that we sleep to clear β-amyloid protein, the protein which has been linked to Alzheimer's disease. It's one reason I'm so passionate about people getting great sleep.
Sleeping in cold conditions has allowed some people with insomnia to sleep faster, raise their metabolism, and feel more refreshed in the morning. All of these can help you have reduced levels of stress during the day, leading you to have more energy to exercise and invest in habits for the positive feedback loop that would get you to sleep the next night.
How do you sleep cold? While cranking the A/C up can help, it can be quite expensive. Using this in combination with fans can cause a decrease in pressure in your room and cool you even more.
Considering really sleeping cold? Consider cooling mattress toppers. When I had no A/C living in California I used this and it worked wonders.