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From my own experiences with my VIPER responder remote that I use everyday, one month is standard on an Alkaline, and about 5-weeks on a rechargeable. The one I keep sitting by my bed, lasts almost two months with a rechargeable. I have the backlight turned on as well as the vibrate, the one in the bedroom has only the beep turned on.
As with any battery powered item, the more it is used the faster it will drain. The installer told me they get about three-to-four weeks on a charge, depending on the settings, and how much use.
I use the Eveready "green" rechargeable for two reasons. They do last longer in electronic devices, and they are a lot cheaper in the long run. (I did some studies with my Garmin 60CSX. I got a steady 8 hours with alkalines with the backlit display on constantly, 14 hours with AA rechargeables...made me a believer.)
Talking with others who have the same type of responder remote, it seems one month is about maximum for alkalines.