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Hyperbolic metamaterials show exceptional optical properties, such as near-perfect broadband absorption, due to their geometrically-engineered optical anisotropy. Many of their proposed applications, such as thermophotovoltaics or radiative cooling, require high-temperature stability. In this work we examine Ag/a-Si multilayers as a model system for the thermal stability of hyperbolic metamaterials. Using a combination of nanotomography, finite element simulations and optical spectroscopy, we map the thermal and optical instability of the metamaterials. Although the thermal instability initiates at 300C, the hyperbolic dispersion persists up to 500C. Direct finite element simulations on tomographical data provide a route to decouple and evaluate interfacial and elastic strain energy contributions to the instability. Depending on stacking order the instability's driving force is either dominated by changes in anisotropic elastic strain energy due thermal expansion mismatch or by minimization of interfacial energy. Our findings open new avenues to understand multilayer instability and pave the way to design hyperbolic metamaterials able to withstand high temperatures.