Frost studied the handwriting of the thirteenth and fourteenth centuries, and on 2 May 1822 he was elected a Fellow of the Society of Antiquaries.
In 1827 he published by subscription Notices relative to the Early History of the Town and Port of Hull; .. with illustrations by Bartholomew Howlett.
[2] In it he claimed to show that the town commonly attributed to have been founded by Edward I (John Leland, William Camden and others), was expanded from an earlier habitation on the same site, named Wyke, which was separate from the place known as Myton to the west.
[3] The work was the subject of a long and flattering critique by Nicholas Harris Nicolas in the Retrospective Review for December 1827.
Frost was president of the society ten times between 1830 and 1855; he served as president of the subscription library for twelve years, between 1827 and 1854, the laws being suspended so that he might occupy the position for five successive years, 1850–4, to enable him to carry into effect his scheme for the amalgamation of the two societies in a building in Albion Street.