Gallium antimonide (GaSb) is a III-V semiconductor of technological interest for low-power, high-mobility field-effect transistors, as well as for mid-wave infrared detectors. In such devices, GaSb interfaces with other III-V semiconductors with different lattice constants that can induce strain in the GaSb layers. Two dominant limiting factors in hot carrier relaxation are the intra-valley and the inter-valley electron-phonon (e-ph) scattering. In GaSb, these are sensitive to the Gamma-L energy ordering, which depend intimately on the strain. Here, we report ab initio calculations of electronic structure, phonon dispersion, e-ph scattering and relaxation times for GaSb as a function of strain. As observed previously for other group IV and III-V semiconductors, our results show strong anisotropy, a strong contribution from LO phonons, and the need to go beyond the deformation potential scattering. For GaSb, the main finding is that a compressive strain between 0.4% and 0.6% converts GaSb from a direct-bandgap semiconductor to an indirect-bandgap semiconductor, with dramatic changes in the competing scattering rates and carrier relaxation times.