Entropic locking of Action Complexity at cosmological singularities

We study the relation between entropy and Action Complexity (AC) for various examples of cosmological singularities in General Relativity. The complexity is defined with respect to the causal domain of dependence of the singular set, and the entropy is evaluated on the boundary of the same causal domain. We find that, contrary to the situation for black hole singularities, the complexity growth near the singularity is controlled by the dynamics of the entropy S, with a characteristic linear relation. This formula is found to apply to singularities with vanishing entropy as well as those with diverging entropy. In obtaining these results it is crucial to take into account the AC expansion counterterm, whose associated length scale must be chosen sufficiently large in order to ensure the expected monotonicity properties of the complexity.