If you're working from MP3, and not doing anything to the audio in the video editing program, then decode it to a regular PCM Wave file (this can be done using any number of tools - lame or a frontend for it is a good option), edit using the Wave file, export the video without audio, compress the video, and then mux the original MP3 back into the final copy. That way no recompression occurs. If you have to edit down the audio, then still decode to PCM Wave, edit the audio in Audacity or something, keep it as PCM, edit the video with that new PCM file, export without audio, and then use the PCM file you edited with to encode an MP3 or AAC file and mux that with the final video encode.
The same is generally true of instances using PCM from the get-go where the video editing program needs the 24fps speedup method because it can't properly handle 23.976fps (like older versions of Premiere). Except in this case you create a new file that's been pitchshifted, edit with that, and perform the fixes on the video before using the original unshifted PCM file to compress the audio to whatever format you need.