In this case, I can't give much advice at all because I don't know what anesthetic agents may have been used. If you know what anesthetic protocol was used, it would help a lot in providing a more specific answer. I am assuming that the depression, drooling and lack of appetite have occurred after the anesthesia. I would be glad to review this information and see if it is possible to help you understand what is happening. Without being able to see and evaluate your dog's condition it is not going to be possible to help much with the diagnosis of the problem but it should be possible to at least give you more specific information on expected clearance times and why a particular anesthetic protocol may have been selected and its effect on the overall situation.
Some anesthetic agents suppress heartrate and others have little effect.
If I induced and maintained anesthesia on isoflurane gas I would expect a more normal heartrate than I would while using a barbiturate anesthetic. There are medications that increase heartrate, counteracting the anesthetic effect. This is why atropine and glycopyrrolate are often used in conjunction with anesthesia. Sorry for the long answer to a seemingly simple question. Inc. Opinions expressed are those of Dr. Richards.