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thats how the OS works. apps dont really completely close per se. the OS only closes apps when it needs more memory.
I am still trying to figure out what the problem is. If i hit pause and return to the home screen it is off for all practical purposes. If I go back into the music player I can select "now playing" or pick something else.
I don't see this as much different then any other phone OS.
If you can catch that downward line in action, you should be able to figure out what process is to blame - the built in battery usage tool would hopefully show it.
I don't know what type of battery life you get, but I can go 2 days with basic use and 1 day if I really use my phone a lot.