Stochastic honeycombs are a random, open cell honeycomb produced through a novel melt-stretching operation. While they have been shown to have excellent mechanical properties under out-of-plane compression, the energy absorption capacity of this cellular material has not yet been examined. The energy absorbed was determined over several of the integration intervals proposed in the literature as a function of density. For two intervals, the relationship between energy and density was linear, and for the other two, the rate of change in volumetric energy absorption capacity with density began to decrease at higher densities. This change happened at a core relative density of 11 %. Additionally, the post-peak collapse mechanisms of four sample sets of varying density were compressed and scanned sequentially through X-ray tomography after preloading to various characteristic strain values. Webs were classified on the basis of their connectivity (bound on both sides or bound on one and free on the other). Unlike conventional honeycombs where all webs undergo the same failure mechanism, the range in geometry of the webs within a given sample led to a range of collapse mechanisms: elastic buckling, plastic buckling, and plastic buckling with fracture. At lower density, all three failure modes could be present in the same sample. At higher density, plastic buckling accompanied by web fracture was the main mode of failure.