I don't think there is one for most airlines. They just have to fit comfortably in the carrier and able to be under the seat. Some may have size restrictions on the size of airline carrier bag. They make them in sizes much larger than what some airlines list as the max size.
9" high I believe is the highest.
I just recently flew with an 8 week old puppy. He was in a sherpa bag that fit under the seat. I flew Continental, and they never asked about the weight of the dog. I think as long as he fits under the seat you will be okay. I did have to show a health certificate, so make sure you have that on hand. I tried to always get an aisle seat, as it is easier to get them in and out. It does get pretty hot under there, so when there was no one sitting next to me I brought him out and set him on the seat next to me and was able to give him a little water. I was lucky as he slept almost the entire trip! Even the TSA agents got big ol' smiles on their faces when we came through security!! Good Luck
Debbie