The only reason I feel the landlord may need to verify employment is because you may be late or not paying your rent as stated in the lease. Otherwise, I do not know why he would need to check. You may want to seek the advice of any attorney.
I guess they could, but you have a lease - right? Is there something in the lease that mentions it or mentions maintaining a job to be able to stay in the rental? Some landlords are more aggressive and demanding than others, but the bottom line is the lease and what it states.