In the case of a rubberband falling into a Schwarzschild blackhole, the work comes from the increasing differential in binding energy between the two ends of the string. This is completely analaguous to the Newtonian case. Binding energy is well defined due to the existence of a time translation symmetry (i.e. timelike Killing field.)
The situation in FRW is slightly different. The rubberband in an FRW metric will only expand if
- The expansion of the of the FRW spacetime is accelerating
- The intitial conditions of the rubberband are such that there is relative kinetic energy between the two ends of the rubberband.
If neither is the case the size of the rubberband will just stay the same, and no work is done.
In case 1, the work comes from the vacuum energy and effectively slows down the acceleration. Instead of a rubber band it is easier to think about an FRW metric filled with an elastic medium. The effect is that you get an FRW solution with a slightly different equation of state (the pressure is a bit larger).
In case 2, the work simply comes from the relative kinetic energy of the opposite ends. This slows down the expansion of the rubber band.