Methods and apparatus to provide vortex cooled secondary flow for a supercritical carbon dioxide pump are disclosed herein. An example thermal management system includes a hot secondary flow extract section fluidly coupled to an outlet of a pump system, a cooled secondary flow section fluidly coupled to the hot secondary flow extract section, the hot secondary flow extract section including a vortex tube to generate cooled secondary air flow from hot secondary flow extract air of the hot secondary flow extract section, and a bearing inlet fluidly coupled to the cooled secondary flow section, the bearing inlet positioned to route the cooled secondary flow to bearings of the pump system.