The Grande Armee was quartered in southern Germany after Austerlitz and remained there until rucking up and going into Saxony and Prussia in September/October. While they were quartered in southern Germany, Napoleon's German allies paid for the Grande Armee's upkeep, which has already been explained.
After Tilsit the Grande Armee was in central Europe but moved through France and into Spain when the first French invasion of Spain failed. What was left in central Europe was 90,000 troops, including the bulk of the heavy cavalry, under Davout. That was fortunate for Napoleon and the French as in early 1809 Austria invaded Bavaria without a declaration of war, just has they had in 1805.