Testing the Equivalence Principle with Strong Lensing Time Delay
  Variations
  Strong lensing time delay measurements provide a valuable and almost
model-independent tool for cosmological investigations. In this work we
recognize that they also carry information on the strength of the gravitational
coupling at the redshift of the lens, and thus could be in principle used to
test the equivalence principle on extragalactic scales. For the case of an
extended lens with a static mass distribution we explicitly derive an
analytical relation between $\dot{G}/G$ and the relative variation of the time
delay. For illustrative purpose, we apply our formula to the light curves of
multiple images of the quasar DES J0408-5354 and simulated ones, which results
in weak constraints on the variation of $\dot{G}/G$ of order $10^{-1}-10^{-2}
\, yr^{-1}$ in the best scenario. Finally, we briefly discuss how those
constraints can be improved in the next future.
