Could you please replase this expression to me?
Stop wasting time! Get to the office on the double.
Does it mean immediately?
"(Do it) on the double" is an idiom that means "(Do it) very quickly".
So, yes, in your example the idea is that the person must come very quickly = immediately.
As far as I know, the idiom comes originally from military usage. Marching "on the double" means the soldiers march at double the speed or twice as fast as usual.
In Damnit's sentence, the sense is basically "without delay" or "immediately" -- i.e. extremely quickly.