As KainPen mentions, Pathfinder doesn't really have a 'train-up' system. When you gain the requisite XP to reach the next level, you automatically* gain the abilities granted by that next level in whatever class you choose to take.
*I say 'automatically' because by RAW there's no implication that training is required to use these new abilities; you achieve the next 'plateau' of experience and they become available:
A character advances in level as soon as he earns enough experience points to do so—typically, this occurs at the end of a game session, when your GM hands out that session's experience point awards.
I would be careful of setting time/gold costs for going up levels.
Read the entire Shackled City book (or take my word for it) - without giving spoilers, there are definitely chapters where there is no time, nothing more than an overnight rest, between chapters but the characters are expected to level up - even in some cases where they are stranded literally in the middle of nowhere with nobody and nothing around but themselves.
So if you establish some weird house rule that it takes two weeks to learn a feat, etc., then during these later chapters you'll have to ignore that house rule or your players will be screwed and unable to level up when they need to - or you'll need to modify the adventure drastically to allow them to spend weeks or months between every chapter even when it really makes no sense at all for them to do so.
I suggest making no such rule.
Instead, let them maybe start a business, or create magic items, or anything else that adds depth to their characters and the campaign without restricting their ability to train up their new levels.
By the way, I am running that AP myself. My PCs are 12th level now. I remember the chapter-break you're talking about. Here's what I did:
I also did some RP foreshadowing and introduced them to the Stormblades. I also nudged them toward building relationships with Fario and Falian (of the Pathfinder Society, venture-captained by Ashstaff - I converted the old "Striders of Farlanghn" into Pathfinder Society), and with Shensen (who is now an NPC cohort), and with Underpriestess Jenya of the church of Iomedae (who is now High-Priestess Jenya - I converted the old 3.0 gods in the book into the gods used in Golarion).
I'm pretty sure the party wizard made a couple magical items in that time too, but they weren't very wealthy at the time so he didn't use the whole time for crafting.
Basically, the PCs built relationships and built a home base during these months - we only spent about a half a session on it and I told them how much time went by and then we started the next chapter.
Why do you need to 'do' anything? In-game time passes in the blink of an eye. Outside of catching up characters who are lagging behind on XP and allowing them to re-stock their goods and\or create new magic items, armor, etc., downtime really shouldn't be necessary.
To echo DM_Blake, if the intention is to require downtime to actively gain new skills, abilities, and such that are normally granted 'automatically' upon gaining a level, I would advise against it, simply because there almost assuredly will be times when the characters may have leveled but they won't have downtime.
The general assumption is that PCs are self-taught, using off-camera time to practice, study, research, and what not. It's glossed over because, well, it's not that interesting.
If you're going to have downtime training, may I suggest using a carrot instead of a stick? Instead of requiring gold and time expenditure to acquire the normal levels and abilities they would receive, give them opportunities to perform tasks that can take large amounts of time that adventurers usually don't have, such as
[list]
Alternately, give them an opportunity to form (or begin to form) a stronghold or other base of operations; later on, this is a good way to RP the recruitment and assignment of followers for any PCs with the Leadership feat.
If the PCs seem underpowered (either XP or treasurewise) for their position in the adventure path, find a module that you can recast into the current game without breaking verisimilitude--or make one up. Think of it like a standalone episode in a book or TV series, an episode that doesn't really mean much in the continuity but offers a chance for the characters to develop a bit.
Finally, you can always just advance the clock. There are campaigns where years go by without any "adventuring". Let them know that you're doing this in case they want to play something out in particular.
I can't recommend Ultimate Campaign enough if you're looking for uses for downtime. In my experience, a lot of GMs brush aside such GM-oriented books as "helpful aids for people who are bad at GMing", but in fact they're quite useful even for experienced GMs who want to broaden their storytelling vocabulary.