With some 12v sockets the dropper is integral, actually part of the socket, and running on it’s rated maximum capacity continuously it will get hot. As Coffeeholic said many are rated at 5amp, and some even less so they will get hot.
If your cool box is ok your probably better off buying a dropper and using a 24v socket, but be sure to get a dropper that is rated more than the cool box. I always had a 12v cool box so I could use it in the car for holidays and weekends 
For several weeks now ive been using a 12v 5.5A coolbox (£30 go outdoors) but over the last week i noticed the plug started to get warm as well as the area around the socket, the plug had not melted but it was starting to discolour so i thought id just change it for a better more robust one 
clever me i thinks, £4, done in 20 mins job done! so i thought.
An hour done the road i went to check it and it had melted 
and as well as the socket was very hot to touch (nearly burnt my hand!!) so when i was back in the yard i popped round to see the
nut- stranglers in the garage and after they checked all the fuses and did a bit of other sparky stuff 
they told me it must be the coolbox? it could be drawing to much power and over heating as if it was the unit the fuses would keep popping or you would have had a cab fire
So in conculsion, i wasted £4, now have no where to store my fresh food/drink and all important milk for brews and nearly went up in a could of smoke and flames and made an jam on the road (but would have probably got me on sally traffic 
)
So the question is do i get another coolbox?
Do i dig deep and go for 24V this time or stick with 12V as they are cheaper and run it through a dropper?
O if so whats the best type to get? and how can i avoid this happening again??
Could be related to the outside temperature. A few weeks ago you might have been keeping the cab cooler with a/c so the cool box didn’t have to work so hard. Now the daytime temperature is lower you are likely keeping the cab warmer so the box works harder and longer while drawing more current.
What’s the rating for the socket, a lot of the 12v ones are rated at 5 amp?
I remember when the fridge looked exactly the same as the roof hatch!
Sounds like a bad connection between the plug and socket raising the resistance and causing it to get hot. Check the surface of the socket and the connection terminal inside are clean.
kayvee:
Sounds like a bad connection between the plug and socket raising the resistance and causing it to get hot. Check the surface of the socket and the connection terminal inside are clean.
I converted mine to run off a Hella socket, much more positive connection.
The stock “■■■ lighter” sockets can be a poor fit and cause the plug to move around, which would cause the problem kayvee mentions above.