Presented here are experiments clarifying how the deformation of cavitation bubbles affects their rebound. Rebound bubbles carry the remaining energy of a bubble following its initial collapse, which dissipates energy mainly through shock waves, jets, and heat. The rebound bubble undergoes its own collapse, generating such violent events anew, which can be even more damaging or effective than at first bubble collapse. However, modeling rebound bubbles is an ongoing challenge because of the lack of knowledge on the exact factors affecting their formation. Here we use single-laser-induced cavitation bubbles and deform them by variable gravity or by a neighboring free surface to quantify the effect of bubble deformation on the rebound bubbles. Within a wide range of deformations, the energy of the rebound bubble follows a logarithmic increase with the bubble's initial dipole deformation, regardless of the origin of this deformation.