Well, it will depend on what you are doing with the phone, to determine how fast the battery is/isn't draining. The more the screen is on and lit, the faster it will drain. That is on top of any apps you have running, even when you aren't actively using them, plus whatever ones you are using.
Games, videos, etc. will drain it even faster.
The heating of the battery could be a fault, but I have to ask 'where are you keeping the phone'? If keeping it say in a pants pocket, then it can heat up more.
If none of the above applies, and it is brand new, either take it back and have them exchange it, or if past the exchange date then contact the manufacturer about a replacement.