A gauge group is the topological group of automorphisms of a principal bundle. For smooth bundles, its classifying space is homotopy equivalent to the moduli space of connections on the bundle. So the cohomology of the classifying space of a gauge group is important in its own right and also for applications. However, the integral cohomology has not been determined in a non-trivial case. I compute the integral cohomology ring of the classifying spaces of gauge groups of principal U(n)-bundles over the 2-sphere by generalizing the operation for free loop spaces, called the free double suspension.