Hot brakes
Lots of rubbish above!
A320 carbon brakes noticably heat up virtually any time they are applied, if he was on a quick turnaround there may have been some residual heat left in them from previous landing and taxy in, although use of the brake fans should have cooled them. With a long taxy out you would need the fans on again.
There is a potential for dirt/crap and even debris to be thrown from the wheels on lift off which can go into the bay and rupture hydraulic lines or worse in the case where a tyre seperates so best to stop then spinning straight away, thats why they are braked, if you dont understand this say CONCORDE several times and you will!
There can be several reasons for leaving the gear down after take off, cooling has already been mentioned, or perhaps even a problem with the sequence, sometime if you think you have a problem that might be gear related and need a return you may not want to run the gear up in case you cant get it down again! Its not always a good idea to explain to ATC exactly what is going on because the reporter with the airband radio will write it up as an a dramatic emegency which isnt good for the companies reputation or image.