My 2011 Edge is doing this exact same thing (but keeping the door open seems to have nothing to do with it, and it never takes more than a few seconds, but it's not a smooth start for sure. Does it every time it's warm/hot out. Cold out is never an issue.).
*** It is not a key ignition. It's push button. ***
They changed the battery and some other sensor awhile back when it was under 100,000 warranty. Out of warranty now. Do you think it could be the same thing with mine?
In TX here, and the summers are VERY hot.
