How do I recover from a cannon rush as Protoss?

Dealing with a cannon rush as Protoss is very different from dealing with it as Zerg. Creep prevents the placement of buildings in a lot of places, so a cannon rush against Zerg is largely used as a contain. This is not true for Protoss.

A number of people have suggested countering, or getting up defense, or expanding, but against a standard cannon rush (8-10 Forge) chances are you don't have time for any of that. They should be finishing their first cannon around the same time you are finishing chrono boosting out your first Zealot. With a very early cannon rush like this, if you do not catch them throwing down that pylon, you will not survive. They've made a huge economic sacrifice and this will be an "all-in" for them.

This is a very scary situation to be in and even at high levels (~1200 Diamond) a fast cannon rush like this a guaranteed win if its not scouted (and still 20-30% effective when it is). I'd recommend the following recourse:

1. Don't panic. Disaster management is a very important skill and if you panic you won't handle this well
2. Don't pull all your workers. This is a classic mistake, as they'll just let them target his Pylon, throw down a new one and cancel the old
3. Kill the Probe. This is usually hard (as you only have probes) and you'll have to pull workers off and micro a surround. Since Probes can build far more than you can destroy this is your number 1 target. This will be especially hard since they can avoid any surround by just clicking on minerals (workers get unit walking while harvesting). Keep attacking them and pull in more Probes as needed.
4. Throw down a Gateway. You should catch them before you'd normally do this so this should be one of your first instincts. Probes can't do enough damage to buildings, you'll need this to clear them out.
5. Don't cut Probe production. While you'll shift your chrono boosts to Zealots (save them while waiting for you gateway), you still need to build your economy. The goal of fighting off this rush is to survive and come out ahead. If you cut Probe production you will not come out ahead.

I highly recommend you practice against this with a friend. Go on Lost Temple and have them sneak a pylon in the back of your base so it overlaps the cliff. They'll follow up by putting their forge on the low ground and the cannons on the high ground (that means 2 probes). If they get out the first two cannons before your Zealot arrives, restart. You want to nip this in the bud.

If your opponent is cannon rushing you, then several things are probably also happening.
The first and most obvious one is that he has little or no unit production. At 150 minerals per cannon, building a gateway back home plus pylon is a no-go, unless your rush has already failed.
A not so evident one is that your opponent has probably been cutting probe production. When I cannon rush I do produce a few probes on the get go, always making sure that I can hit the first 100 and then 150 for the pylon / forge. Thus, they will probably have a lesser amount of income.
And finally, cannons back at home. Cannon rushing is cheese. And as all cheese, it is thought to take the opponent by surprise and unprepared. Building a pylon and at least one cannon back at home means 250 minerals less towards forward cannons. That's almost 2 less cannons to attack with. In my experience, if there is a cannon rush, there are no cannons back at home unless the rush failed and the opponent is trying to fall back to economy play (which imho, cannot happen unless you allow it).
Considering these, I would say that good PvP responses to cannon rushing would be:

1) Take whatever units you have and go for their probes. No money = no cannons. If you still have an economy going, killing off the cannons at your base is a only a matter of amassing some troops (assuming your opponent bm's and doesn't gg).
2) Expand if you have the minerals to. Abandon your base, relocate elsewhere in the map and make sure you don't get cannon rushed again t.t . You will be ahead in economy and units. Make sure that they don't expo to your original spot. Go for their probes, again, no money = no cannons or units.

I believe that the key concept when being cannon rush is realizing what is not happening. The usual response to such a situation is "OMG I'm dead", and seldomly do we think "Well, they can't move their cannons, they've invested a HUGE amount of money on this attack, and right now they have nothing but probes back home". Not trying to say that it is easy, but the key with such a situation is not falling into desperation.

Hope this was useful.
Regards.

Build a few cannons between his cannons and your vulnerable buildings in order to prevent him from advancing. Then just ignore the cannons and attack.