Barbarossa had turned over the government of the empire to his son, and it seems highly likely that he intended to stay in the east indefinitely. He wouldn't need to go home soon as Richard and Philip had to do.
That's interesting, is there anything that suggests he was going to establish himself to rule and then establish a cadet house from one of his sons? (i.e. Frederick of Swabia)
Considering the sheer size of the Crusader forces with his troops there, and the potential for him to provide singular leadership and arbitray between Richard and Phillip, it could make the Third Crusade much more than the reclamation of Acre, but the potential to found an Empire of Outremer (at least, that is what I'd expect if Frederick was intending to stay). Restoring Jerusalem is all well and good, but if we've retained the Germans, and French, then you could well have a united force not just liberate Jerusalem (regardless of the issues between Guy and Conrad, which need to be resolved) but campaign to Damascus and Egypt.
In fact, a greater Third Crusade would make sense if it was about recapturing Jerusalem, securing Damascus to protect against the North and East, and then move down to Egypt. Egypt, Jerusalem and the rest of the Crusader States under a Barbarossa-led Empire of Outremer sounds pretty epic (and sensible) to me.
What WERE his plans? Do we have any ideas?