In this talk, I shall consider the convolutional double copy for BRST and anti-BRST covariant formulations of gravitational and gauge theories. I shall give a general BRST and anti-BRST invariant formulation of linearised $\mathcal{N}=0$ supergravity using superspace methods and show how this may be obtained from the square of linearised Yang-Mills theories. I shall then demonstrate this relation for the Schwarzschild black hole and the ten-dimensional black string solution as two concrete examples.