It depends on the context. Hopefully the sentences below will help you: 1) He lived in hope of getting a better job. 2) He sent CVs out to multiple employers in the hope of getting a better job. When we do something with a specific purpose then you may often see the expression "in the hope of," as per the second sentence. But it would sound very odd to insert a "the," in the first sentence.

Hi Estella, The correct phrase should be either: In hopes of or In the hope of. Both are in this case acceptable. In hope of is fine grammatically, but it is not the way it is normally used. Hope this helps. Aleks