As checkboard noted, the battery is largely inconsequential in light of the potential damage to the APU. If the battery voltage is low, slower spool up times for the start will occur, and potentially hotter starts may occur.
A low battery results in longer starter engagement times, and potentially the inability to spool the engine fast enough.
A battery which has been drained will take greater current when receiving a charge after the start, which can result in higher batter temperatures. In the case of NiCad batteries the potential for a thermal runaway exists...which definitely does damage to the battery...as well as the surrounding airframe.