Why did Doug stop working for Frank ??

He never stopped working for Frank like you may think. After he got bashed in the head by Rachel and was hospitalized, Frank didn't want him coming back to full-time work during his recovery. Primarily because Frank was afraid Doug had A) become sloppy, and B) wouldn't be able to handle the workload. To prove himself, Doug joined up with Dunbar so he could prove he could do the work, and also get inside info from the Dunbar campaign and take it to Frank, which would prove his continuing loyalty. Hence, his showing up with that journal and subsequently burning the pages that would have otherwise painted Claire in a bad light for having had several abortions.

He was hospitalized and recovering. Frank most likely thought he was now week and a liability so he was shutting him out.

At first Frank wouldnt let him back in and then it appeared Doug was going to work for the enemy to get back at Frank. But in reality Doug is like a Loyal dog to frank and just wanted to get back to his feet.

Doug was on mission to prove himself again and take his spot as Frank's right hand.