does it say at which phase?

if you're using the latest firmware it will show what's loading at the bottom part of the screen.

Is your BB icon a thick one, or a thin one?

hmm. that means it has failed to load.

Have you tried downloading the phone's firmware and autoload it?

Its like a phone reset, but this process reloads your phone's firmware, and you lose EVERYTHING inside the phone.

most welcome.

just remember, DO NOT load beta firmware. Just load official firmware.

Current official firmwares are for 10.3.1
Now there are a lot of 10.3.2 betas.. once you load that you can never go back (your phone will brick)

hold out, don't upgrade to 10.3.2 yet. its not even officially out.

your PIN may have been whitelisted for 10.3.2 beta, but trust me, you don't want to upgrade to that YET.
When your phone is brand new (as in auto reloaded), it will force all cores (dual core) to run to optimise settings. Usually there's a cooling down period.
You should make sure to:

a) activate BBM
b) activate your email accounts
c) connect to a wifi
Normally when your phone is newly reloaded it tends to ask for things such as your login details. If you fail to provide it, it will continue talking to servers as if its still trying to send data (although its lack of data), and thus processes are run, and your phone heats up.

that's 10.3.1.1779, the latest for 10.3.1

that's good.
no need to stop it.
if its 10.3.2 it won't be from there, at least not yet.
That's where you click to get official updates.
It says its updating to 10.3.1.1779. You should get it.
if it says 10.3.2 BETA, don't.