Electrical engineer here, and I've wired up entire harnesses for a 1952 MG TD and the engine compartment of a 1982 240 Wagon that I drove for 30 solid years.
Your cooler sounds like the problem, and needs to be in its own socket and on its own fuse. (Stuff with motors and compressors and fans are the big current sucking monsters, the consumer electronics chargers are mostly negligible in their current demands.)
You don't want a bigger fuse for the same wiring, as the designer put in appropriate wire sizes for 20Amps max, and the 20Amp fuse is protecting the wires and the rest of the car from overloads. If you want a bigger fuse, you have to upgrade the wiring first.
This is not so hard to implement, as each 12 Volt socket has its own wire running to the fuse. Read through, as you have to go exploring first, and then buy a terminal and some wire at Radio Shack or a hardware store to do the job. I'm going to write this up with extra warning and advice for even complete novices, because tiny electrical mistakes can be expensive in a car with multiple computers.
0) Pull the negative battery cable. (I wish there was also a sure-fire way to also "disable all airbags", like in the early 1990s 740 wagons. Anyone know if there is a fail-safe disabling method?) Anyway, always pull the negative battery cable before doing electrical work, and slide the cable into a zip-lock bag to keep it from touching all the metal that lurks nearby.
1 ) Pull the fuse ("11 C/13" on my v70, wiring diagram attached, it is in Group 37 if you have a different model/year).
2 ) Get at the two Violet wires that run to the "cold side" of the fuse (the side without 12 Volts on it when fuse is removed) which may require removal of various plastic and bits of trim.
3 ) Snip one of the two wires with wire cutters.
4 ) Snip the wire on the cold side of an unused fuse position for an accessory you'll never have.
5 ) Solder the 12 Volt socket wire you cut away from fuse C/13 to the cold side of that unused fuse.
6 ) Insulate your solder joint with electrical tape well, and tape the bit of wire you snipped away from, as there is now a "bare wire" at the end of where you cut.
7 ) Insert a 20 Amp fuse into the unused fuse position, and document the change on the fuse panel cover and in a note in your owner's manual and service record.
8 ) You can now test both sockets to make sure that your wiring/soldering job was good, as each should now only get power with the appropriate fuse in place. But you aren't quite done yet...
9 ) Both 12 Volt sockets share the same (solid black) ground wire. They need individual ground wires if you want to increase the load they can carry, don't they? In my wiring diagram, they connect together at junction point "53/406" which is a wiring harness point that is amidships, likely right under the center armrest where the rear socket is located. You'll have to pull it apart, something I've never done, so look elsewhere for step-by-step on doing that, and post a link in this thread for those who come along later.
10 ) You need to separate the grounds, so that each socket has its own wire to a solid ground. So, you have to find a grounding point. The easy way to do this is to probe around with a volt-ohm meter with one lead connected to a wire attached to a known "frame ground", such as the jump-starting point in the engine compartment. Keep checking bolts, rubbing each with some steel wool to get a good clean surface, until you find one that is a true ground. Cut the black wire that comes from the socket, and solder on a longer length of wire of the same or larger size, and solder it to a ring terminal appropriate for the bolt size and your wire size. They look like this: http://www.grainger.com/Grainger/3M-Ring-Terminal-4X310
. Don't just crimp the terminal if you want to draw some serious current from the socket, solder it too. Slide he ring terminal on your ground bolt, and install the bolt.
10A ) It may be easier to find a good ground in the radio/ECC area where the front socket lives, and it is not so hard to pull the environmental control unit (WITH BATTERY DISCONNECTED!) to do this. There are several excellent sets of instructions on this with photos. So, you'd cut the black wire from the 12 volt socket, and attach it to a good ground you find behind the dash, which leaves only the rear socket using the ground point "53/406".
11 ) Again, insulate your soldering and the end of the wire your cut with electrical tape.
There! Now you have two 12 Volt sockets, each with a 20Amp fuse, each with its own ground wire, and you can plug the cooler into the back one, and the consumer electronics into the front one.
I don't have the socket in the rear cargo compartment, but its power is supplied by the "REM", so I would not mess with it, as it is already a stand-alone socket with its own power and ground. (Anyone care to comment on why Volvo has a computer-controlled 12 Volt socket in the cargo bay? Sounds to my like a set-up for a scene from "2001":
"Open the pod bay door, Hal."
"I can't do that Dave, I'm installed in a Volvo V70 Wagon, and you can't open the pod bay door with a dead battery."
"OK, how about the cooler in the back - can you keep the fish I caught cold?"
"Sure Dave, I can do that, I am a HAL-9000, and I am very good friends with the REM."
"REM? You know the members of the group 'REM'?"
"No, Dave, the box of circuity in the back of the car - that's an REM".