$RO(C_p\times C_p)$-graded cohomology of universal spaces and the coefficient ring

We compute the $RO(C_p\times C_p)$-graded Bredon cohomology of equivariant universal and classifying spaces associated to families of subgroups, with coefficients in the constant Mackey functor $\underset{‾}{{\mathbb{F}}_p}$. An explicit description of the resulting coefficient ring, including its multiplicative structure, is obtained. These computations are then applied to the study of lifts of cohomology operations via the Bredon cohomology of equivariant complex projective spaces.