[Nightedge]

Could I use his son's? (not simply his son)

-This hope will cost __ freedom.

Dictionaries have sentences:

-That one mistake almost cost him his life.
-A late penalty cost United the game.
-Such behaviour could cost you your place in the team.

without the possessive s. But the original sounds OK.

[susan53]

I don't really understand the context so I'll use a different example:
This action cost .... life.

Context : David's father Peter was a bad driver. He took a bend in the road at 90 mph and crashed .

a. This action cost David his life.
b. This action cost Peter's son his life.
c. This action cost his son his life
d. This action cost his life.
e. This action cost him his life
f. This action cost Peter his son's life.
g. This action cost his son's life.
h. This action cost his son life[/i]

To cost XXX something = XXX loses something
To cost something = XXX is lost.

Notice that d and e are ambiguous - it's not clear if we're talking about Peter or David. But the problem with (H) is that the sentence This action cost his son life is grammatically correct but difficult to understand and therefore unlikely to be used. (b) This action cost his son his life. or This action cost his son's life are much clearer and more likely.

[Nightedge]

Excellent answer. Thank you.