Double barrel guns were updated in the reprinting of Ultimate Equipment.
In the double-barreled pistol description, in the first sentence, change everything after the semicolon to “each barrel can be fired independently as separate attacks, or both can be fired at once as a standard action (the attack action).” Followed by the standard "If both barrels are shot at once, they must both target the same creature or object, and the pistol becomes wildly inaccurate, imparting a –4 penalty on each shot."
Even with the guncrafting rules he probably shouldn't have too many guns seeing as each Double-barreled Pistol is costing him 875gp. Also his attack bonus is gonna be abysmal with TWF and I'm guessing Deadly Aim as those together would impose a -6 penalty alone, and if he was using the old double barrel feature on top of that he'd have a -10 penalty total on all of his attack rolls.
Also he hasn't even gotten to the Dex to damage part of being a Gunslinger so he's entirely reliant on Deadly Aim and Up Close and Deadly(which costs grit per shot) for damage. So with TWF or a standard action double barrel shot you're only looking at 2d8+8(or 10) or if he wants to really kill the target 2d8+2d6+8(or 10) for damage, while he shoots with a -6 penalty. If he doesn't have Precise Shot or an ally is providing cover things get even worse for his chances to hit.
Finally don't forget his pistols have a short range increment for hitting Touch AC meaning foes will have no problem getting to him, and that the guns can misfire.
Ability Scores: Str 10; Dex 19 (16+2+1); Con 14; Int 14; Wis 14; Cha 10
Feats (1) Two-Weapon Fighting, (1B) Quick Draw; (3) Point-Blank Shot; (4B) Deadly Aim
Gear: (6,000gp wealth expected).
A masterwork double-barreled pistol is 2,050gp, and per the gunsmithing feat this is 1 day of work per 1,000gp of market value, and costs half the market value. So about 2 days and 1,025gp per gun. Assuming he carries 4 (for 2 rounds of combat) that's 4,100gp of his wealth, leaving enough for a +1 chain shirt for decent AC.
BAB +4
Within 20ft
Single Shot: +8 (1d8+5, vs touch AC)
Standard Action: +4/+4 (1d8+5, vs touch AC)
Full Attack: +2/+2/+2/+2 (1d8+5, vs touch AC)
Plus buffs. Inspire Courage in particular is sickening for the gunslinger.
For comparison, a Fighter with a masterwork composite longbow and archery feats (Weapon Focus, Point-Blank Shot, Precise Shot, Rapid Shot, Deadly Aim, Weapon Specialization) has a range increment of 110ft, and within 30ft would be looking at:
Standard Action: +9 (1d8+9)
Full Attack: +7/+7 (1d8+9)
Add to that his defenses should look like:
AC 20 (Base 10, +4 Dex, +5 armor, +1 Nimble)
HP 34
Fort +6; Ref +8; Will +3
At 5th level they gain an extra +4 to damage with each shot, and can take Rapid Shot and make 2 extra guns to add another 2 shots per round on top of the above. i.e. +1/+1/+1/+1/+1/+1 (1d8+9)
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The big issue is touch AC, and the fact that it largely doesn't scale. For any given CR of creature the average touch AC is about 12 or so. though it depends what you're fighting. Against tiny and small sized creatures, your gunslinger PC will struggle as their touch AC is high. Against anything reliant on natural armor (most big things), your gunslinger will wreck face, and this trend will only continue as they get higher level.
In theory, the powerful nature of firearms and their special rule of targeting touch AC within the first increment is balanced by the trickiness of reloading and the misfire rules, both of which are negated by using multiple weapons and the Quick Draw feat. You don't even need to take Rapid Reload or use Paper Cartridges (which increase misfire chance) because you are rarely bothering to reload in combat.
Some Suggestions:
1) Raise the Dex and Dodge component of your creatures AC, but try to keep the total AC the same. This might not seem like a big deal, but given the gunslinger is tanking their attack bonus into the floor, every point of touch AC will help bring their damage closer towards the rest of the party.
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2) Use reinforcements and hidden foes. Don't do this every fight, but having some of the monster's allies out of sight when combat starts, or kicking the door in during Round 3 would encourage your gunslinger player to be slightly more careful with his ammunition, as reloading all four guns takes time that most combat doesn't allow, or at the very least it gives the rest of the party something to fight and feel useful against.
3) Watch the dice. When someone is rolling between 4 and 6 attacks a round, rolls of 1-2 (or 1-3 with paper cartridges) will happen, which not only are misfires (and thus cannot hit) but render that weapon broken for all subsequent shots. As long as the gunslinger has Grit, they can restore them as a standard action - not an issue if combat ends after 1-2 rounds, but in a protracted combat it will make a huge difference. Also note the broken condition applies to the weapon, not just the single barrel.
4) Fire and Water. Black powder is ruined by exposure to water (though cartridges and a powder horn protect it), so if he's carrying around loaded (and not with cartridges) firearms all the time and you want to ruin his day, drench that cat in water, and force him to reload. Or set him on fire. Unless he has magic items that protect his powder from such, any natural 1 on a Ref save against fire means his gear is damaged by the effect = A Very Bad Day when his fighting style involves carrying a large stockpile of unstable explosives.
5 Damage Reduction. Specifically DR/magic is your friend here. Most PCs can afford a magic weapon by level 4-5 fairly easily, but the gunslinger is not going to be able to afford the 4-6 magical weapons needed to make all his attacks count as magical, forcing him to instead rely on magical ammunition, which goes away at an alarming rate. So throwing in creatures with DR/magic (such as Gargoyles) which the other characters can bypass, but the gunslinger struggles against is a neat way of leveling the playing field and avoiding the the pony's One Trick from trumping everything.
Just some food for thought :)
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To Raynulf list I would add: keep a tally of how much shots he fire (or even better, have some other guy keep that tally,as you, as the GM would be busy doing other things) and check that he pay for them (either in money or in time crafting them).
Even if he make its ammunition by himself, at 1,1 GP/shot the cost of firing 8 shots in each encounter is a drain on resources.
Well, he also has a situation where he makes extra powder/guns to sell on the market, thus keeping his resources up.
also, he posted this to me today, saying that this page doesn't mention the standard action: http://paizo.com/pathfinderRPG/prd/ultimateCombat/combat/firearms.html#pist ol
Tell him to use a site which updates their information. Paizo's website uses the same text as in the original books, so you'd need to really search through it to find all relevant information and erratas.
d20pfsrd
Has (almost) everything. Easy to navigate through hotlinks. Updates their information.
Archives of Nethys
More accurate than d20, but not as easily navigated. Updates their information.
There have been FAQs and Erratas to patch the "dublicate all my attacks in a full-round" exploit. Several of them. Your player may not have done it intentially, but that is now against the rules.
And the ammunition and guns he crafts with the Gunsmithing feat are sold at half the market price, so he can't earn any money from doing so.