Anyway I also noticed that with apn and background data enabled, the phone is warmer and eats battery power. This isn't the case when apn is enabled but background data is disabled. So I reckon it might be some software issue...
Yes, back to the topic...
That's natural! Google Sync is keeping connection all the time with background data on (and auto sync on, but I suppose it is on on you phone).
Think of it - I have Google Talk installed on my laptop and it notifies me of new e-mails. My phone is always the first to know there's a new message. That means that polling interval of Gmail sync in the phone is smaller than that in Google Talk. I think it may even be real-time. That's why the phone is always online and using data and the battery so heavily.