We have studied the thermal conductivity $kappa$ on single crystalline samples of the antiferromagnetic monolayer cuprates R$_2$CuO$_4$ with R = La, Pr, Nd, Sm, Eu, and Gd. For a heat current within the CuO$_2$ planes, i.e. for $kappa_{ab}$ we find high-temperature anomalies around 250 K in all samples. In contrast, the thermal conductivity $kappa_c$ perpendicular to the CuO$_2$ planes, which we measured for R = La, Pr, and Gd, shows a conventional temperature dependence as expected for a purely phononic thermal conductivity. This qualitative anisotropy of $kappa_i$ and the anomalous temperature dependence of $kappa_{ab}$ give evidence for a significant magnetic contribution $kappa_{mag}$ to the heat transport within the CuO$_2$ planes. Our results suggest, that a large magnetic contribution to the heat current is a common feature of single-layer cuprates. We find that $kappa_{mag}$ is hardly affected by structural instabilities, whereas already weak charge carrier doping causes a strong suppression of $kappa_{mag}$.
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