Is my landlord allowed to pop in my apartment whenever he pleases?

No, your landlord may not do this. When you lease, you gain possession of the unit your landlord may own it, but he does not possess it while you are leasing. You are also given a right to "peaceful enjoyment," which is the right to use the residence without undue or unreasonable disruption. 
The landlord has certain limited rights to enter your unit without notice, for an emergency fire, large leak, etc. with at least 24 hours notice, at reasonable time, and not too often, for maintenance, to inspect its condition, or to show for prospective buyers or tenants.
However, apart from the above, the landlord may not "pop into" your unit at anytime. If he does, he is violating the implied covenant of quiet enjoyment and you may have grounds to terminate your lease without penalty if he keeps doing this after you provide written notice to him send it some way you can prove deliver to stop. If you are thinking of going this route, be sure to carefully document whenever he does come in you may also wish to consider putting in a "nanny cam" aimed at the front door, to record when he comes in especially is he does so when you are not home.