Magnetic anisotropy is anticipated to govern the formation of exotic spin textures reported recently in cubic chiral magnets, like low-temperature tilted conical and skyrmion lattice (SkL) states and metastable SkLs with various lattice geometry. Motivated by these findings, we quantified the cubic anisotropy in a series of CoZnMn-type cubic chiral magnets. We found that the strength of anisotropy is highly enhanced towards low temperatures. Moreover, not only the magnitude but also the character of cubic anisotropy drastically varies upon changing the Co/Mn ratio. We correlate these changes with temperature- and composition-induced variations of the helical modulation vectors, deformations of skyrmions, structural rearrangements of the metastable SkLs and the large enhancement of Gilbert damping. Similar studies on magnetic anisotropy are required for the quantitative, unified description of the known stable and metastable modulated spin textures and for the systematic exploration of novel ones in this large class of skyrmion hosts.