We investigate how changes in the boundary metric affect the shape of AdS black holes. Most of our work is analytic and based on the AdS C-metric. Both asymptotically hyperbolic and compact black holes are studied. It has recently been shown that the AdS C-metric contains configurations of highly deformed black holes, and we show that these deformations are usually the result of similar deformations of the boundary metric. However, quite surprisingly, we also find cases where the horizon is insensitive to certain large changes in the boundary geometry. This motivates the search for a new family of black hole solutions with the same boundary geometry in which the horizon does respond to the changes in the boundary. We numerically construct these solutions and we (numerically) explore how the horizon response to boundary deformations depends on temperature.