> Are they saying that an application running continuously in the background uses the same amount of battery as an application that's not running at all?
A single application running in the background probably uses only a small amount of extra power compared to the notification service alone.
However, a couple dozen applications -- all running in the background in order to enable notifications -- probably consume a lot more power than a single service.
There's also a wide variety of possible implementations of a notification service, and not all of them will necessarily be as power efficient as what Telegram-FOSS does.
A single application running in the background probably uses only a small amount of extra power compared to the notification service alone.
However, a couple dozen applications -- all running in the background in order to enable notifications -- probably consume a lot more power than a single service.