Aims. The nucleus of comet 67P/Churyumov-Gerasimenko (67P) is ice rich and shows evidence of morphologies driven by sublimation processes. In particular, the fine-particle deposits (FPD) that cover its surface reveal depressions of many tens of meters. A detailed analysis of these morphologies and of the properties of the fine-particle deposits could help to constrain the amount of water ice in the subsurface. The aim of this paper is to characterize these depressions and constrain their formation and evolution by a detailed quantitative study of their morphometry and thermal environment, and through a comparison with similar morphologies on Earth and Mars. Methods. To study the cometary depressions, we constructed a database of morphometrical parameters for 131 depressions on 67P, 50 alases on Earth, and 200 scalloped depressions on Mars. We measured these parameters manually with the ArcGIS software. We used the images of the Narrow Angle Camera for 67P (down to a resolution of 1 m pixel−1), of the USGC-Digital Orthophoto Quadrangle for the Earth (5 m pixel−1), and of the High Resolution Imaging Science Experiment for Mars (25 cm pixel−1). We also used the digital elevation models associated with these images. In addition, we performed a study of the thermal environment of each depression on 67P to estimate the maximum temperature, the total erosion, and the total energy received from the Sun during one revolution. The morphometrical and thermal environment parameters were then used together for a comparative study of the depressions on 67P, the Earth, and Mars. Results. Our results show that depressions on 67P are geological analogs to alases on Earth and scalloped depressions on Mars; we call them cometary thermokarst depressions. They have been formed by a thermokarst process (i.e., permafrost thawing that causes a subsidence) triggered by water-ice sublimation. The cometary thermokarst depressions on 67P are distributed throughout the nucleus, without preferential distribution for a particular hemisphere or region. They are exclusively located in the fine-particle deposit layer, which is a few meters thick and is thought to have to contain a volume fraction of water ice of about 50% for the subsidence to occur. The formation and evolution of the depressions is driven by the Sun; the side facing the dominant insulation is the preferential direction of erosion. Our results contribute to a better understanding of the periglacial system on comet 67P and of the erosion processes on the nucleus.