Car won't start if the battery has been on for ~5 minutes

I have a 2006 Honda Accord, and my car battery is not too old (1-2 years). The car starts fine, I don’t normally have issues with it. However if I turn off my engine and sit in my car with the battery on for not even five minutes, my car won’t start (won’t turn over). If I then let it rest for another 10 minutes, it starts fine again.

Even though my car doesn’t start, the headlights, interior lights, radio, etc., all work fine. I live in cold climate, but it’s not that cold here yet (30s and 40s F).

Could I need a new battery replacement, even though my battery is not very old? Or could this be indicative of something else?