Burthen is an archaic variant of burden. It was common in English as recently as the late 19th century, but it is mostly absent from the language today. It was often used as a nautical term denoting the tonnage of a ship, as in phrases like “a schooner of 200 tons burthen.” This sense of burthen outlasted all the others, but it is now obsolete as well.
The absence of officers and privates from their duty under various pretexts, while receiving pay at great expense and burthen to the Government, makes it necessary that efficient measures be taken to enforce their return to duty. [New York Times (1862)]
[C]heerily have ye made me tread the path of life with all the burthens of it (except its cares) upon my back. [The Life and Opinions of Tristram Shandy, Gentleman, Laurence Sterne (1759)]