Parametrized cohomology of the classifying space for $\mathbb{Z}/2$-line bundles

The Thom isomorphism fails for equivariant vector bundles in $RO(G)$-graded cohomology, even for $G=\mathbb{Z}/2$. Costenoble--Waner developed a parametrized equivariant cohomology theory incorporating Thom isomorphisms for all vector bundles. This cohomology theory extends the usual representation grading $RO(G)$ to $RO(\Pi_G B)$, representations of the equivariant fundamental groupoid. We compute the parametrized cohomology of the classifying space for real $\mathbb{Z}/2$-line bundles, $B_{C_2}O(1)$. This is joint work with Agnès Beaudry, Chloe Lewis, Sabrina Pauli, and Elizabeth Tatum.