Mechanosensation of cells is an important prerequisite for cellular function, e.g., in the context of cell migration, tissue organization, and morphogenesis. An important mechanochemical transducer is the actin cytoskeleton. In fact, previous studies have shown that actin cross-linkers such as α-actinin-4 exhibit mechanosensitive properties in their binding dynamics to actin polymers. However, to date, a quantitative analysis of tension-dependent binding dynamics in live cells is lacking. Here, we present a, to our knowledge, new technique that allows us to quantitatively characterize the dependence of cross-linking lifetime of actin cross-linkers on mechanical tension in the actin cortex of live cells. We use an approach that combines parallel plate confinement of round cells, fluorescence recovery after photobleaching, and a mathematical mean-field model of cross-linker binding. We apply our approach to the actin cross-linker α-actinin-4 and show that the cross-linking time of α-actinin-4 homodimers increases approximately twofold within the cellular range of cortical mechanical tension, rendering α-actinin-4 a catch bond in physiological tension ranges.