In this paper we consider the semilinear damped wave problem of the form {(α(t)ut)t−β(t)Δu+γ(t)ut+δ(t)u=β(t)f(u),x∈Ω,t>τ,u(x,t)=0,x∈∂Ω,t⩾τ,u(x,τ)=uτ(x),ut(x,τ)=vτ(x),x∈Ω, where Ω is a bounded smooth domain in RN, N⩾3, τ∈R, f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α,β,γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space.