A new method for performing density-based topology optimization for Stokes flow is presented. It differs from previous approaches in the way the underlying mixed integer problem is relaxed. We extend the existing theory for the generalized Stokes equations, which allows us to choose BV or fractional order Sobolev spaces for the density. We investigate the arising optimization problems concerning existence of solutions, differentiability, and convergence of relaxed solutions towards solutions of the original problem. We motivate a localized fractional order Sobolev norm as an approximation of the BV -norm for ±1-valued functions and discuss its discretization for piecewise constant finite elements. Building on these theoretical findings, we present our numerical realization and show computational results.