I just replaced my thermostat about a three weeks ago. It ran fine for about a week and a half, and now I’m starting to have a little bit of a rough start and misfiring when cold. It goes away after 30-45 seconds then is fine. It definitely sounds like it’s misfiring, but I have no check engine light or codes. I just put a battery in it a couple of days ago. And I did close and wires in the summertime less than 1000 miles ago. I saw somewhere in a form people say to change the oil. that seems a little bit odd to me, but I’m gonna do that this week. Does anybody have any ideas after the oil change? I’m probably gonna do a camp position sensor, but ideally, I’d like to not throw money at it and hope something eventually fixes it.
Does sound like the cam position sensor, but would read the codes first. If replacing, get an OEM one! 
With a hot engine at idle, check the 4 fuel trims (2 short term and 2 long term). Let us know what they are.