We study thermoosmosis as the physical basis for creating a micro- or nanoscale engine. As a model system, we consider concentric cylinders in which an annular cavity has been created. The outer cylinder, which acts as a stator, comprises two halves composed of different materials. The inner cylinder forms a rotor that is driven by thermoosmotic flow within the annular cavity etched into the stator and filled with a nonionic liquid. The circular thermoosmotic flow is established within the enclosed cavity in response to an external temperature gradient directed transverse to the cylindrical axis. The circular flow induces rotation of the rotor through hydrodynamic friction at the rotor surface. The velocity and direction of the rotation depends on the magnitude of both the temperature gradient and difference in physicochemical properties of the two stator materials. The thermoosmotic engine can be used to drive a mechanical load in micro- and nanoscale devices. The drive load characteristic of the engine is expressed as the dependence of the angular velocity on the load torque.