Let X be a space and write LX for its free loop space equipped with the action of the circle group T given by dilation. The equivariant cohomology H^*(LX_hT;Z/p) is a module over H^*(BT;Z/p). We give a computation of this module when X=CP^r for any positive integer r and any prime number p. The computation does not use the fact that CP^r is formal, nor does it use the Jones isomorphism and negative cyclic homology.