As DO at a 135 operation, I've worked with the FAA on these issues and it's always been smooth. One of the biggest concerns is that flight crews need to follow maintenance procedures correctly. If they aborted due to a mechanical abnormality such as a warning light, the FAA wants to make sure the underlying issue is addressed before the airplane flies. This is oftentimes as simple as a phone call to Maintenance from the runup pad to discuss what happened, but the FAA doesn't want pilots to shrug it off and take to the air on attempt #2 with a broken airplane.
Also, on a related note, we do a lot of in-aircraft training and checking at my operation. Obviously we have to train and check over aborted takeoffs. Lots of parties were annoyed when this mandatory reporting of aborted takeoffs came to be and our operation was generating a ton of reports due to training. We changed our technique to request "high speed taxis" down runways or "short delays at the beginning of our takeoff" with Tower so as to not constitute an abort for the sake of paperwork.