The actin cortex is an active biopolymer network underneath the plasma membrane at the periphery of mammalian cells. It is a major regulator of cell shape through the generation of active cortical tension. In addition, the cortex constitutes a mechanical shield that protects the cell during mechanical agitation. Cortical mechanics is tightly controlled by the presence of actin cross-linking proteins, that dynamically bind and unbind actin filaments. Cross-linker actin bonds are weak non-covalent bonds whose bond lifetime is likely affected by mechanical tension in the actin cortex making cortical composition inherently mechanosensitive. Here, we present a quantitative study of changes in cortex composition and turnover dynamics upon short-lived peaks in active and passive mechanical tension in mitotic HeLa cells. Our findings disclose a twofold mechanical reinforcement strategy of the cortex upon tension peaks entailing i) a direct catch-bond mechanosensitivity of cross-linkers filamin and $\alpha$-actinin and ii) an indirect cortical mechanosensitivity that triggers actin cortex reinforcement via enhanced polymerization of actin. We thereby disclose a `molecular safety belt' mechanism that protects the cortex from injury upon mechanical challenges.