Rough idle after start-up
 

Ok so I have searched this site and the net for a cause/solution but don't think it was ever really answered. If I missed it on here I am sorry. So I have a 2011 JKU with 13,000 miles on it and about 3 weeks ago the engine light came on. Jeep was running fine, so I just ran down the road and had advanced auto run the code and clear it. The code was a small evap leak. So I had them clear the code and it has not came back since.
The last few days the jeep has been running rough on start-up. It seems to be idling bad , and shakes the jeep. After about 5 minutes or a short drive it goes away and runs as good as new. I can start it back up and its fine. It only happens during the 1st start of the day. About 2 days ago I also had a message saying gascap, but I got out and it was tight. It also cleared and has not came back on.
Does anybody have any ideas on what this could be? Could an EVAP issue/gas cap cause the rough idle/shaking on start-up? I am also thinking about having the battery checked as maybe its weak and dying. Since it stops after 5 minutes, I don't think it is anything serious, or anything motor related. Anyway if anybody has any ideas I would be glad to hear them. Thanks