It's likely because late night/early morning is cooler temperature, and this drops the battery voltage enough to trigger the sensor.
tl;dr: the battery, in this case, was in a room that is continuously controlled for humidity, temperature, and in this case: light.
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The system usage should also be pretty static: as our house guest was home and presumably tripping the motion sensors all day. (Yes: even though the system was unarmed _and unactivated_, our motion sensors still appeared to be _sensing._ -- It's purely coincidence, but this ordeal started right around when the Internet was in an uproar over PRISM & the NSA.)
Retiring to our unprotected bedroom dwellings for the evening would've only _decreased_ the system's draw, if it had any impact at all.