Cope came from a cadet branch of a Northamptonshire family which first represented that county in 1397. His great-grandfather acquired the Oxfordshire manor of Hanwell in 1498.
Cope was sheriff of Oxfordshire at the general election of 1604, and therefore ineligible to sit in Parliament. He yielded his former seat at Banbury to his eldest son, Sir William Cope*, and presumably supported the return of his brother-in-law John Doyley* as knight of the shire. Following the end of his shrievalty he managed to secure a seat for himself, being returned for the county at a by-election to replace (Sir) Lawrence Tanfield*, who had been appointed a judge on 13 Jan. 1606. He took up his seat by 12 Feb., when he was added to the committee for a conference on supply.
In a supply debate on 14 Mar. Cope declared that ‘because many good bills are in the House, and more will come’, grievances should be redressed before there was any grant of subsidy.
The main business of the third session was the king’s project for the Union with Scotland. On 24 Nov. 1606 Cope moved that the Instrument of the Union should be returned to the Lords, whereupon he was ordered to attend a conference on the following day.
When Speaker Phelips fell ill late in March 1607, Cope was one of those appointed to consider precedents for procedure during his absence.
Outside Parliament, in June 1607, Cope joined with his brother and others in what Chamberlain later called ‘a great bargain with the king’ for the purchase of chantry lands and parsonages to the value of £32,000.
When the fourth session met in 1610, Cope was named to a committee to consider two bills against pluralism and non-residence (19 Feb. 1610).
In 1610 Cope received a lease of woods in Whittlewood forest and a grant of the manor of Bruern, Oxfordshire.
Cope was returned to his final Parliament in 1614, when he served as senior knight of the shire for Oxfordshire. At the age of 74, he was one of the oldest Members in the House. On 5 May he opposed putting supply to the question, saying that although ‘the king’s part might carry it by voices, yet it would not be so honourable for the king to have one negative voice’.
On 9 June, two days after the dissolution of the Addled Parliament, Cope made his will, ‘being sick in body’. Although he made generous provision for his three younger sons, he reportedly left debts of over £20,000, some of which were incurred on behalf of his brother.
