We propose a novel class of temperature-tunable semiconductor metamaterials that exhibit negative refraction in the terahertz spectral range. These metamaterials are based on doped semiconductor superlattices with ultrathin barriers of about 1 nm thickness. Due to the tunnel transparency of the barriers, layers of the superlattice cannot be considered as isolated and, therefore, the classical homogenization approach is inapplicable. We develop a theory of quantum homogenization which is based on the Kubo formula for conductivity. The proposed approach takes into account the wave functions of the carriers, their distribution function and energy spectrum. We show that the components of the dielectric tensor of the semiconductor metamaterial can be efficiently manipulated by external temperature and a topological transition from the dielectric to hyperbolic regime of metamaterial can be observed at room temperature. Using a GaAs/AlGaAs superlattice slab as an example, we provide a numerical simulation of an experiment which shows that the topological transition can be observed in the reflection spectrum from the slab.