Don't they just? I have one with a rechargeable battery which needs to be charged about once a week when it's generally only on for about 45 minutes a day. It does seem odd that when we are all trying to cut our energy consumption that we should be introducing a broadcasting technology that uses much more energy than traditional analogue wireless.
If you were to wind up a "Wind up" radio for a couple of minutes it would play an FM station for about an hour. The same radio would play a DAB station for about five minutes. The processing power needed to decode the signal is enormous and accounts for the low battery life.
DAB and Digital TV are very economical on the bandwidth required for a station but at the expense of power needed to decode the signal.