It's the same XC time as the career pilot program, just less of it. You will likely be paired up with career pilots for the ASAP program, for that matter. One pilot is under the hood as pilot flying, the other is safety pilot and handles navigation and radios. Both of you log PIC, as specified in the FARs.
I flew with a couple of ASAP guys during my XCs, and noticed that both had sub-par instrument skills, to the point of being uncomfortable in extended IMC. Not to generalize at all-other than that they did great, and we had some great flights.