There were many "Houses of the Rising Sun" - from way back. A few samples:

Worcester [England] Herald (May 5, 1798):

"The well-accustomed PUBLIC HOUSE, called THE RISING SUN, situate in Droitwich, in the county of Worcester."

Bury and Norwich Post (Sept. 2, 1801):

"The Public-house, called THE RISING SUN...in Bury St. Edmonds."

Richmond (Va.) Inquirer (Oct. 7, 1807): Adjoining the Rising Sun Tavern."

Long-Island Star (Brooklyn, N.Y.) (May 17, 1827):

"These are the same gang that broke into a house near the Rising Sun, on the Germantown road."