The Kost surname is derived from the personal names Kosty and Kost, which are pet forms of the male given name Konstanty, which is the Polish form of the name Constantine. The name Constantine is a derivative of the Latin name Constantinus, which means steadfast and faithful. This name was borne by the first Christian ruler of the Roman Empire, Constantine the Great (280-337). Constantine rebuilt the city named Byzantium, renamed it Constantinople and made it the capital city of the Eastern Roman Empire. Constantinople is the modern city of Istanbul, Turkey.
In the United States, the name Kost is the 11,357th most popular surname with an estimated 2,487 people with that name. 
^ Filby, P. William, Meyer, Mary K., Passenger and immigration lists index : a guide to published arrival records of about 500,000 passengers who came to the United States and Canada in the seventeenth, eighteenth, and nineteenth centuries. 1982-1985 Cumulated Supplements in Four Volumes Detroit, Mich. : Gale Research Co., 1985, Print (ISBN 0-8103-1795-8)