He spends pretty much most of the time in there (When not in the dining carriage/kitchen because he loves food). Mostly for one of three reasons:

1) General training be it sword or hand to hand
2) Practicing performances (He has some juggling balls so now and again he likes to pratice)
3) Pokemon time. He has like five of them now. They need to train and such to.

The most likely thing that would work is the juggling, Allen can get easily focused and really into it.