A cooling system in which an electronic device or other component is cooled by using one or more solid sources of liquid vapor (such as polymeric absorbents, hydrates or desiccants that desorb water at comparatively low temperature) in conjunction with one or more high-temperature vapor sorbents or desiccants that effectively transfer heat from the component to the fluid in the wellbore. Depending on the wellbore temperature, desiccants are provided that release water at various high regeneration temperatures such as molecular sieve (220-250° C.), potassium carbonate (300° C.), magnesium oxide (800° C.) and calcium oxide (1000° C.). A solid water source is provided using a water-absorbent polymer, such as sodium polyacrylate. Heat transfer is controlled in part by a check valve selected to release water vapor at a selected vapor pressure.