My 5D MK III is sitting next to me with my 70-200mm lens mounted, I haven't used it since Monday. I reached over and half pressed the shutter, and it came on in 1/2 second. The battery will last for months if left in hibernation with the power on, but I use it enough to recharge the battery every week or two at a minimum.
If yours is not coming right on, do a camera reset and see if it still happens. Make sure you don't have any third party battery grip or the like on it.