Abstract In this work we consider a spacial kind of spacetime called an anti-de Sitter accelerating black hole. This is a kind of black hole that contains a stringlike singularity along polar axes attached to the black hole that accelerates it. By using a string as a probe, we study the properties of the complexity growth of black holes following the complexity–action duality. We find that the growth in the Einstein–Hilbert action is independent of the acceleration but the string probe detects the effects of acceleration.