Born on 6 January 1746 at Elmly near Huddersfield, he was the third son of Matthew Topham (died 1773), vicar of Withernwick and Mappleton in Yorkshire, and of his wife Ann, daughter of Henry Willcock of Thornton in Craven.
Topham died without issue at Cheltenham on 19 August 1803, and was buried in Gloucester Cathedral, where a marble monument was erected to him in the nave.
With Philip Morant, Richard Blyke, and Thomas Astle he collected and arranged the Rotuli Parliamentorum from 1278 to 1503, published for the Record Commission, to which he was secretary, in six volumes between 1767 and 1777.
In 1775 he edited Francis Gregor's translation of Sir John Fortescue's De Laudibus Legum Angliæ and (with Richard Blyke) Sir John Glanvill's Reports of certain Cases … determined … in Parliament in the twenty-first and twenty-second years of James I, to which he prefixed "an historical account of the ancient right of determining cases upon controverted elections".
In 1781 the Society of Antiquaries published a tract by him, A Description of an Antient Picture in Windsor Castle representing the Embarkation of King Henry VIII at Dover, May 31, 1520 (London), and in 1787 he contributed Observations on the Wardrobe Accounts of the twenty-eighth year of King Edward I [1299–1300] to the Liber Quotidianus Contrarotulatoris Garderobæ published by the Society under his direction.