Coefficients in $\mathbb Z$?
– Fernando MuroJun 20 '13 at 17:36

There is paper by Loday "Free loop space and homology". It establishes isomorphism between cohomology of free loop space $LX$ and Hochschild homology of an algebra of singular cochains $S^*X$. Maybe this will help.
– Sasha PatotskiJun 20 '13 at 17:37

@Fernando Muro, Yes. But \mathbb{Q} coefficeints is also fine. @Sasha Patotski, thanks, I will have a look.
– Murat SaglamJun 21 '13 at 14:49

Do you mean for each component of the free loop space? The free loop space $LM$ has $p$ components for the lens space $M=L(p,q)$.
– Somnath BasuAug 30 '13 at 14:26