Let $X$ be a space and write $LX$ for its free loop space equipped with the action of the circle group $\mathbb{T}$ given by dilation. We compute the equivariant cohomology $H^*(LX_{h\mathbb{T}};\mathbb{Z} /p)$ as a module over $H^* (B\mathbb{T} ;\mathbb{Z} /p)$ when $X=\mathbb{C}\mathrm{P}^r$ for any positive integer $r$ and any prime number $p$. The computation implies that the associated mod $p$ Serre spectral sequence collapses from the $E_3$-page.