"In the hope of" vs. "with the hope of"

A quick comparison in COCA shows that in the hope is more commonly used.

Even ODO has this expression used in its example sentence to demonstrate the usage of hope.

hope

feeling of expectation and desire for a particular thing to happen:

he looked through her belongings in the hope of coming across some information

Though in the hope of and with the hope of are pretty much interchangeable.

Please take note that in hopes of can have a different meaning, as suggested by the New Oxford American Dictionary. (I don't find it in other dictionaries though.)

in hopes of

with the aim of: I lay on a towel in the park in hopes of getting a tan.

Both are grammatical. There seems to be little difference in meaning, but a detailed corpus search might show that they were used in different contexts. What corpus evidence does show is that in the hope is more popular than with the hope. It occurs nearly four times as much in the Corpus of Contemporary American English, and nearly eighteen times as much in the British National Corpus.

All variants of hope in this context are somewhat informal, and therefore don't really match the tone of OP's sentence. So I'd suggest using "with the aim/intention of".

In less formal contexts, I would slightly favour "in the hope of". But it really is just a stylistic choice between several alternatives, all of which would normally be semantically identical...

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