We present sufficient conditions to obtain a generalized ( φ,𝔇) –pullback attractor for evolution processes on time–dependent phase spaces, where φ is a given decay function and 𝔇 is a given universe. We deal specifically with the case in which φ has either exponential or polynomial decay, the universe is the one of uniformly bounded families, and apply the abstract results to a non–autonomous wave equation with time–dependent propagation velocity.