We introduce the method of double pole QCD sum rule which is basically a fit with two exponentials of the correlation function, where we can extract the masses and decay constants of mesons as a function of the Borel mass. We apply this method to study the mesons: $\rho(1S,2S)$, $\psi(1S,2S)$, $\Upsilon(1S,2S)$ and $\psi_t(1S,2S)$. The experimental masses are obtained in two distinct Borel windows with the formation of a curious local maximum to $\psi(1S)$ and $\Upsilon(1S)$. The $\rho$ meson decay constant is closer to the experimental semileptonic decay constant than di-electron decay constant, on the other hand the values of $\psi(1S,2S)$ decay constants are in agreement with the experimental di-electron decay constant, if we consider two distinct Borel windows. In the cases of $\rho(1S,2S)$ and $\Upsilon(1S,2S)$ the decay constants show discrepancies with the experimental di-electron decay values, which can be explained as the effect of QED vacuum polarization. We also present predictions for the toponiuns masses $\psi_t(1S,2S)$ of m(1S)=357 GeV and m(2S)=374 GeV.