Thermal maintenance costs 15% of global energy. The concept of an "energy-free thermostat" was accordingly proposed to change environmental temperature gradients, which can maintain a certain space at constant temperature without the need to consume additional energy according to a temperature-trapping theory [Shen et al., Phys. Rev. Lett. 117, 055501 (2016)]. Here we develop the theory by introducing thermoelectric effects, and we then propose a "negative-energy thermostat" that generates electricity associated with energy-free maintenance of a constant ambient temperature. The thermostat further helps to design a different thermoelectric cloak: for changing ambient temperature gradients, its central region has an approximately constant temperature and potential while simultaneously generating thermoelectromotive forces. Our finite-element simulations confirm the effects predicted by the theory developed, and we also discuss the possibility of experimental demonstration with natural materials. This work not only has relevance for expanding the temperature-trapping theory by achieving thermoelectric conversion synergistically but also offers a different method to manipulate coupled fields in electrothermotics; it also gives hints on how to control other coupled fields in photothermotics, photoelectrics, electromagnetics, thermomagnetics, and thermomechanics.