Even if (or when, as I agree a falling out is inevitable) the HRE and France drift apart in the immediate years following Otto's downfall Philip Augustus will have a greater freedom to act in the short term.
The falling out may end up being quicker : Philippe Auguste's support of Philipp of Swabia wasn't exactly stellar by 1208, probably due to the fact Welfs were going to be defeated and an agreement between the KoR and the Pope about to be issued.
I don't think that it would be as immediatly tense it was with Otto IV, of course, but I don't think that an Honestaufen HRE would be good news for Capetians and, more importantly, they'd know that : I'd even expect some tensions raising about Flanders or french infuence over Barbant.
Louis VIII invaded England and claimed the Crown during the Barons War of 1214-1216
Which quickly died off with the death of John Lackland, and his son being a much more better alternative than he was, or that the son of the guy that put out Anglo-Norman nobles out of Normandy and beneficied from a growing power, was.
The issue was never about military success, but how politically it could be reached. I don't see how an Honestaufen HRE could change that in the latest.
@LSCatalina I take issue with the idea that French expansion into Burgundy/Arles will be taken equaninimously.
Where in frozen hell did I wrote that?
What I wrote was litterally that Capetian policy when it come to an Honestaufen HRE in the first part of the XIIIth century would be ambivalent, rather than the alliance you suggested.
Anyhow : Capetians weren't really interested in imperial Burgundy, not before they managed to get a real hold on their part of the rhodanian corridor which isn't bound to happen as quickly as IOTL. It depends a lot from how the crusade goes, and while a meridional victory isn't that likely (I shouldn't say that, collaborating to a meridional victory TL) without particularily favourable circumstances, a Montfort victory could end up with a large principality that would abort the mediterranean policy of Capetians for the time being.
If IOTL, you didn't have a clear Capetial policy along the Rhone before the XIVth century, I doubt it would appear with an Honestaufen-led HRE in the early part of the XIIIth.
The Staufers, based out of Swabia and Sicily, were fundamentally southerly in their outlook- not for nothing did Frederick I become one of the few Emperors to receive coronation in Arles.
Burgundy was at this point imperial mostly in name, as hinted by the Great Southern War in the XIIth century.
Frederic II's coronation in Arles is more of a symbolical gesture (symbolical doesn't mean void, of course) targeted less at the kingdom of Burgundy than a point made to Italy, following roughly the same policy there that Otto IV did except that Frederick given up on naming imperial agents and named the count of Toulouse/marquis of Provence as his representent which was the admission that a Swabian emperor had few direct powers there.
It's not for nothing that Frederick II was the last emperor to really attempt claiming Burgundy's suzerainty. This beaten horse had long since died.
French expansionism in that direction would not be met passively IMHO.
Again, which French expansionism in imperial Burgundy in the early XIIIth? Even the annexation of Lyon took decades and it wasn't before the XIVth that it became significant.
Assuming that the situation goes more or less as IOTL in the region, you'd probably end as a matrimonial diplomacy to neutralise Provence, which would be far less targeting imperials (that had no real power in the region since litterally centuries), than Aragon.
For similar reason I have difficulty seeing Central Italy passively leaving The Imperial aegis.
I'm afraid I don't see what's really confusing in my post : I said it would be a
non-decisive agreement, about giving
part of imperial influence.
There's no much difficulty there, mostly because it was part of what Otto IV did, what Philip of Swabia probably negociated with Innocent III (while probably without surrendering everything Otto IV did, especially when it come to imperial suzerainty.
I didn't specifically mentioned the Adriatic region as well (and I think we both agree that Philip giving away everything up to Ancona is not going to happen), even if I think it would be part of negociations, I'd rather see the IOTL proposal (that never went anywhere due to Philipp's death) to give away some influence on western Central Italy, aka Tuscany.
How long such agreement will last is anybody's guess, but it would certainly not last forever : hence why I precised
uneasy peace.
As the Emperors power waned it is likely that Frederick would expand his influence and perhaps his kingdom northwards as an Imperial vicar, official or otherwise.
It's possible, but I doubt it would happen easily IMO. Dynastic solidarity played a lot in XIIIth politics (as hinted by the Plantagenet/Welf alliance), and even if Frederick II is at odds with Philip, you'd need a full fledged war between Honestaufen Sicily and HRE to really get rid (temporarily) of the threat of dynastical encirclement for what concerns Rome.