You could, provided you're willing to modify them accordingly. Juggle the XP awards so that they don't rocket past the entire AP "ahead of level".
It can get the players complacent if you laregly run it "as is", saving serious "work" in the early chapters for the BBEGs sprinkled throughout. Once they are 'caught up' to your comfy point, then you should be able to run the AP "as is" after that comfy point.
Well, unless you're borderline nutters like I am and you upgun the entire AP ... but that's another story.
AP= Adventure Path
BBEG = Big Bad Evil Guy or a major villian
I would not suggest starting an AP at book 1 with a party of 7th level. The reasons are two-fold: you'd need to scale up every statblock and you'd need to change the layout to accomidate high-level spellcasters. (There are some lowlevel AP plothooks and quest premises that are negated by things like Dispel Magic or Teleport.)
However, you can start an AP at book 2 or 3 without too much hassle. In particular:
Rise of the Runelords: starting at book 2 would be a bit tricky due to its ties to book 1, but you can start at book 3 very easily. Just find someone to set you on a quest to check out Fort Rannick.
Curse of the Crimson Throne: haven't read it personally (hope to play someday!), but from what I gather one thing really leads to another, so this wouldn't work.
Second Darkness: starting at book 3 would be easy. One of the "flaws" of this AP is that books 1&2 are so disconnected from books 3-6. Make that flaw an advantage! Just have the PCs get buddy-buddy with some elves.
Legacy of Fire: starting at book 2 would be pretty easy: just task the group with taking out the Carrion King or otherwise securing the northern mountains. Starting at book 3 directly would be trickier, as the party needs certain things to make it work. This campaign (while awesome, my 2nd favorite actually) has problems in that it is very tightly themed (Arabic) and that might not mesh with what you've got going on already.
Council of Theives: like CotCT, I hope to play someday, so I really can't say.
Kingmaker: you either do this beast whole-hog or you don't do it at all. Would lose a lot if you tried to jump in higher level. Sorry.
Serpent's Skull: you could start at book 2 without much effort (just make sure to get them a map), or you could start at book 3, but that would be rather abrupt (and a poor way to introduce APs if you care to read my review of said book).
Carrion Crown: I've only read the first two books so far. It seems pretty modular so far. Probably can jump right in. Though like LoF, it's very heavily themed (gothic horroc), and that might be a no-go for you.
Hope that was helpful!!
Coucil of thieves could work skipping books 1 and 2. Book 2 is pretty fun though. The first part is more about introducing NPCs and leveling you up to 3rd. Not much story but more setting the scene. Book 2 while fun could also be skipped as well having the NPCs doing the work and present the Players with the hook for Part 3. That way you would be starting at 5th. Skipping the 3rd book would be difficult.
I'm running Kingmaker as a "sequel" to Council of Thieves, with the PCs starting at the equivalent of 12th level in Chapter 1 of Kingmaker. They'll finish Chapter 3 at 15th and Chapter 4 at 16th. The nice thing about Kingmaker is its semi-realistic nature in terms of encounters. There are things "out there" that are across all the possibilities. Quite a few are underwhelming - instant vaporization for the high level characters. Others are overwhelmingly nasty. Compare the "as written" possibility of running into a small pack of trolls or owlbears in Chapter 1.
It can be done, it just depends upon how much effort you can afford to put into it.
There are excellent guidelines above for using "as written" materials to drop 7th level characters into. Typically 7th is the beginning range of an AP's 3rd Chapter.
Run with what strikes your fancy!