Unlike the wide-ranging dynamic control of electrical conductivity, there does not exist an analogous ability to tune thermal conductivity by means of electric potential. The traditional picture assumes that atoms inserted into a material's lattice act purely as a source of scattering for thermal carriers, which can only reduce thermal conductivity. In contrast, here we show that the electrochemical control of oxygen and proton concentration in an oxide provides a new ability to bi-directionally control thermal conductivity. On electrochemically oxygenating the brownmillerite SrCoO2.5 to the perovskite SrCoO3-delta, the thermal conductivity increases by a factor of 2.5, whereas protonating it to form hydrogenated SrCoO2.5 effectively reduces the thermal conductivity by a factor of four. This bi-directional tuning of thermal conductivity across a nearly 10 +/- 4-fold range at room temperature is achieved by using ionic liquid gating to trigger the 'tri-state' phase transitions in a single device. We elucidated the effects of these anionic and cationic species, and the resultant changes in lattice constants and lattice symmetry on thermal conductivity by combining chemical and structural information from X-ray absorption spectroscopy with thermoreflectance thermal conductivity measurements and ab initio calculations. This ability to control multiple ion types, multiple phase transitions and electronic conductivity that spans metallic through to insulating behaviour in oxides by electrical means provides a new framework for tuning thermal transport over a wide range. Unlike dynamic control of electrical conductivity, tuning thermal conductivity by means of electric potential is challenging. Electrochemically induced phase transition control of oxygen and proton concentration in a SrCoOx oxide provides a way to tune bi-directionally thermal conductivity.