REBCO coated conductors (CCs) have the potential to widen considerably the application areas of superconductivity. Quench protection of REBCO-based devices represents one of the major obstacles to this expansion. Thermal conductivity data are mandatory ingredients for quench simulation studies. In particular, the transverse thermal conductivity (κT) of the CC plays a key role in case of quenches in superconducting magnets. In the present work, we report κT of thermally stabilized REBCO CCs produced by different manufacturers. We have measured κT on single CCs rather than on stacks of soldered tapes, this excludes spurious contributions in the measurements. We have found that the presence of the stabilizer at the tape edges in Cu-electroplated CCs increases the overall κT. In the absence of this effect, the overall thermal resistance is dominated by the substrate.