  We compute the $\text{SO}(n+1)$-equivariant mod $2$ Borel cohomology of the
free iterated loop space $Z^{S^n}$ when $n$ is odd and $Z$ is a product of mod
$2$ Eilenberg Mac Lane spaces. When $n=1$, this recovers Ottosen and
B\"okstedt's computation for the free loop space. The highlight of our
computation is a construction of cohomology classes using an
$\text{O}(n)$-equivariant evaluation map and a pushforward map. We then
reinterpret our computation as giving a presentation of the zeroth derived
functor of the Borel cohomology of $Z^{S^n}$ for arbitrary $Z$. We also include
an appendix where we give formulas for computing the zeroth derived functor of
the cohomology of mapping spaces, and study the dependence of such derived
functors on the Steenrod operations.
