The half-lives of isotopes around the $N=82$ shell closure are an important ingredient in astrophysical simulations and strongly influence the magnitude of the second $r$-process abundance peak in the $Asim130$ region. The most neutron-rich $N=82$ nuclei are not accessible to the current generation of radioactive beam facilities and $r$-process simulations must therefore rely on calculations of the half-lives of the isotopes involved. Half-life measurements of the experimentally accessible nuclei in this region are important in order to benchmark these calculations. The half-life of $^{130}$Cd is particularly important as it is used to tune the Gamow-Teller quenching in shell-model calculations for the $beta$ decay of other nuclei in this region. In this work, the GRIFFIN $gamma$-ray spectrometer at the TRIUMF-ISAC facility was used to measure the half-life of $^{130}_{~48}$Cd$_{82}$ to be $T_{1/2}= 126(4)$ ms. In addition, the half-lives of the three $beta$ decaying states of $^{131}_{~49}$In$_{82}$ were measured to be $T_{1/2}(1/2^-)=328(15)$ ms, $T_{1/2}(9/2^+)=265(8)$ ms, and $T_{1/2}(21/2^+)=323(50)$ ms, respectively, providing an important benchmark for half-life calculations in this region.
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