He was educated at St Paul's School and Trinity College, Cambridge, where he matriculated in 1706.
His elder brothers, Charles and Henry, both succeeded to the earldom before him but died unmarried.
Coote succeeded in his turn on 27 March 1720 and ascended to the Irish House of Lords.
In 1741 he stood for Parliament again at Hedon in Yorkshire, and was initially declared defeated.
They had only one child, Charles (c. 1725–1802), who succeeded to the earldom on Mountrath's death in 1744, but who died unmarried, the title thereby becoming extinct.