Why my phone does not need to update the password? I changed my gmail password a day ago, but my phone does not require me to put in the new one. I can still access my gmail with the old password. Why?

Phone is rooted, sim unlocked , running android 2.3
Your phone doesn't store the password, it has some kind of session key.  I would have expected it to be invalidated when you changed the password, but apparently Google doesn't do that.  You'd have to ask them why.