CHDK does not increase battery consumption.
But... it has recently been deduced that Canon firmware of a570 and a590 (and very likely all other models too) have a firmware upgrade safety mode in their battery low detection: before Canon firmware enters a firmware upgrade, it may shut down with batteries half full because it tries to make sure the update itself will not fail due to low batteries.
Problem is, CHDK is always started as a firmware upgrade. CHDK doesn't actually upgrade firmware, but its start mechanism is the same as with Canon firmware upgrades, which is why this safety functionality is triggered.
Logically thinking this feature affects dual AA cell cameras the most because their supply voltage is lowest.
The only solution I know for now is to use a good quality, new (in respect of manufacturing date and recharge cycles) batteries only. In my experience with my dual AA a570 a good pair of NiMH will be discharged almost fully even when running CHDK, but 18 months later the same pair (still in good condition for other uses) may not even let the camera start in REC mode after a full recharge!
And when the camera won't start but you need to shoot, disable CHDK autoboot (by unlocking your SD card) and use your camera without CHDK. It will work for a while longer.