Arran was the son of Sir Arthur Gore, 2nd Baronet, and Elizabeth Annesley, and was educated at Trinity College, Dublin.
In 1727 he was elected to the Irish House of Commons for Donegal Borough, a seat he held until 1758.
This was the third occasion that the title Earl of Arran had been created in the Peerage of Ireland.
[2] His sister Anne married John Browne, 1st Earl of Altamont in December 1729 and had ten children.
He died in April 1773 and was succeeded in his titles by his eldest son Arthur.