Terrence Howard -- Pay Up!! Accused of Stiffing Landlord After Eviction

Terrence Howard ain't got to go home, but he's gotta get the hell out of a NYC apartment -- 'cause the landlord has evicted him, and is now suing the actor for back rent ... TMZ has learned.

According to the docs, filed in Manhattan, Howard's lease on the West Broadway apartment ended back in November, 2012 -- but he kept living there until at least May of this year ... when he was formally evicted.

The landlord is now suing Howard for an unspecified amount of unpaid rent ... as well as legal fees.