Thermal conductivity (κ) plays an essential role in functional devices. It is advantageous to design materials where one can tune κ in a wide range according to its function: single-crystals and nanowires of anatase polymorph of titanium dioxide, broadly used in applications ranging from photovoltaics, reflective coatings to memristors, have been synthesized in large quantities. Here we identify a new, strong diffusion mechanism of heat by polaronic structures due to oxygen vacancies, which considerably influences both the absolute value and the temperature dependence of κ. The additional decrease of κ is achieved in anatase nanowires organized into foam, where porosity and the quasi-one-dimensional size-effect dramatically hinder the propagation of heat, resulting in an extremely low κ = 0.014 W/Km at room-temperature. Doping this anatase foam could herald promising applications, in particular in thermoelectricity.