Internally structured self-assembled nanospheres, cubosomes, are formed from a semicrystalline block copolymer, poly(ethylene oxide)-block-poly(octadecyl methacrylate) (PEO39-b-PODMA(17)), in aqueous dispersion. The PODMA block provides them with a temperature-responsive structure and morphology. Using cryo-electron tomography, we show that at room temperature these internally bicontinuous aggregates undergo an unprecedented order-disorder transition of the microphase-separated domains that is accompanied by a change in the overall aggregate morphology. This allows switching between spheres with ordered bicontinuous internal structures at temperatures below the transition temperature and more planar oblate spheroids with a disordered microphase-separated state above the transition temperature. The bicontinuous structures offer a number of possibilities for application as templates, e.g., for biomimetic mineralization or polymerization. Furthermore, the unique nature of the thermal transition observed for this system offers up considerable possibilities for their application as temperature-controlled release vessels.