We present results on the rest-frame $H$-band luminosity functions (LF) of red sequence galaxies in seven clusters at 1.0 < z < 1.3 from the Gemini Observations of Galaxies in Rich Early Environments Survey (GOGREEN). Using deep GMOS-z and IRAC $3.6 mu$m imaging, we identify red sequence galaxies and measure their LFs down to $M_{H} sim M_{H}^{*} + (2.0 - 3.0)$. By stacking the entire sample, we derive a shallow faint end slope of $ alpha sim -0.35^{+0.15}_{-0.15} $ and $ M_{H}^{*} sim -23.52^{+0.15}_{-0.17} $, suggesting that there is a deficit of faint red sequence galaxies in clusters at high redshift. By comparing the stacked red sequence LF of our sample with a sample of clusters at z~0.6, we find an evolution in the faint end of the red sequence over the ~2.6 Gyr between the two samples, with the mean faint end red sequence luminosity growing by more than a factor of two. The faint-to-luminous ratio of our sample ($0.78^{+0.19}_{-0.15}$) is consistent with the trend of decreasing ratio with increasing redshift as proposed in previous studies. A comparison with the field shows that the faint-to-luminous ratios in clusters are consistent with the field at z~1.15 and exhibit a stronger redshift dependence. Our results support the picture that the build up of the faint red sequence galaxies occurs gradually over time and suggest that faint cluster galaxies, similar to bright cluster galaxies, experience the quenching effect induced by environment already at z~1.15.
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