The gas-solid budget of carbon in protoplanetary disks is related to the composition of the cores and atmospheres of the planets forming in them. The principal gas-phase carbon carriers CO, C0, and C+ can now be observed regularly in disks. The gas-phase carbon abundance in disks has thus far not been well characterized observationally. We obtain new constraints on the C/H ratio in a large sample of disks, and compile an overview of the strength of Ci and warm CO emission. We carried out a survey of the CO 6-5 line and the Ci 1-0 and 2-1 lines towards 37 disks with the APEX telescope, and supplemented it with Cii data from the literature. The data are interpreted using a grid of models produced with the DALI disk code. Gas-phase carbon abundance reductions of a factor of 5-10 or more can be identified robustly based on CO and Ci detections, assuming reasonable constraints on other parameters.