Time dependent density functional theory (TDDFT) is widely used to calculate excitation energies of molecules. It is however not amenable to local-type approximate exchange-correlation functionals which are commonly used in density functional theory. Using the current instead of the density as the basic variable (time dependent current density functional theory (TDCDFT)) remedies this problem. We derive scaling relations for TDCDFT similar to relations previously given for TDDFT: we obtain the behavior of the exchange-correlation potential and kernel under simultaneous scaling of space and the electron-electron coupling constant. This can be used to verify that approximate functionals satisfy the scaling or to obtain the exchange-correlation potential at various densities by simply knowing the potential as a function of the coupling constant. The scaling relation can also give the ground-state exchange-correlation energy through the adiabatic connection formula.