It's ok-it probably just means he or she is hungry. A trick I learned from my nurse when my 1st child was born is to mix a TINY amount of Gerber's Rice Cereal with either breast milk or formula, whichever you're using. The mixture should still be liquid, just slightly thickened by the cereal. Put it in a bottle and make a cross-cut in a nipple, so the baby can drink the thickened liquid. Feed your baby that bottle at midnight or so, and it should hold him or her for at least a couple of extra hours, so you can get some sleep! This was pretty common knowledge back in the day, but now it seems to be out of style. Remember that babies haven't changed, only doctors have. I have 5 grandchildren; it worked just fine when my kids were babies, and none of them developed any food allergies or illnesses. But check with your pediatrician to be sure.
In my view, it is normal as children have different eating habits and some are just insatiable. the best thing to do is consult a doctor to advice you on what to do. Since your child is still young, you will be advised on what you can give your child to make him full and the periods in which he/she is fed.
Don't feed your one month old rice cereal in a bottle!! He is way to young for that. If you think he might be eating to much observe how he acts during and after a feeding. Give him 2oz or so then try to burp him. If he continues to suck well keep feeding him. Does he turn his head to the side or away from the bottle? He may be done or need to burp. If he is spitting up a lot this is also a good indication he's had too much. Overall, I think if a baby is hungry feed him :) if your not sure, try meeting all his other needs i.e. diapers, simulation etc... Congratulations on being a new mom!