Abstract The synthesis of wafer-scale two-dimensional amorphous carbon monolayers has been recently demonstrated. This material presents useful properties when integrated as coating of metals, semiconductors or magnetic materials, such as enabling efficient atomic layer deposition and hence fostering the development of ultracompact technologies. Here we propose a characterization of how the structural degree of amorphousness of such carbon membranes could be controlled by the crystal growth temperature. We also identify how energy is dissipated in this material by a systematic analysis of emerging vibrational modes whose localization increases with the loss of spatial symmetries, resulting in a tunable thermal conductivity varying by more than two orders of magnitude. Our simulations provide some recipe to design most suitable ‘amorphous graphene’ based on the target applications such as ultrathin heat spreaders, energy harvesters or insulating thermal barriers.