Terahertz optics offer the potential to image through objects that are opaque for visible wavelengths and provide unique spectroscopic signatures for a variety of materials and quantum processes. However, the resolution of THz images suffers from the long wavelength of THz light compared to visible. Hyperbolic metamaterials provide a possible solution through the creation of superresolving lenses and offer greater flexibility in effective refractive index than can be provided by natural materials. Most hyperbolic metamaterials function in a narrow bandwidth due to their resonant nature. In search of a broadband material, we simulate a temperature-tunable hyperbolic metamaterial composed of a multilayer stack of alternating layers of high-density polyethylene (HDPE) and indium antimonide (InSb). At a single temperature, negative effective medium permittivity is found over a small bandwidth of 0.09 THz, but by tuning over a 40°C temperature range, the bandwidth is increased dramatically to 1.0 THz. Furthermore, we compute the transmission and negative refraction through the multilayer stack and simulate the imaging properties of curved hyperlens stacks using slits as test objects, achieving resolutions as small as 20 μm at 130 μm wavelength, far below the half-wavelength diffraction limit.