The integrated response function in phase-ordering systems with scalar, vector, conserved, and nonconserved order parameter is studied at various space dimensionalities. Assuming scaling of the aging contribution chi(ag)(t,t(w))=t(w)(chi)(-a)chi(t/t(w)) we obtain, by numerical simulations and analytical arguments, the phenomenological formula describing the dimensionality dependence of a(chi) in all cases considered. The primary result is that a(chi) vanishes continuously as d approaches the lower critical dimensionality d(L). This implies that (i) the existence of a nontrivial fluctuation dissipation relation and (ii) the failure of the connection between statics and dynamics are generic features of phase ordering at d(L).