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Multilayered metal-dielectric nanostructures display both a strong plasmonic behavior and hyperbolic optical dispersion. The latter is responsible for the appearance of two separated radiative and nonradiative channels in the extinction spectrum of these structures. This unique property can open plenty of opportunities toward the development of multifunctional systems that simultaneously can behave as optimal scatterers and absorbers at different wavelengths, an important feature to achieve multiscale control of light-matter interactions in different spectral regions for different types of applications, such as optical computing or detection of thermal radiation. Nevertheless, the temperature dependence of the optical properties of these multilayered systems has never been investigated. In this work, we study how radiative and nonradiative processes in hyperbolic meta-antennas can probe temperature changes of the surrounding medium. We show that, while radiative processes are essentially not affected by a change in the external temperature, the nonradiative ones are strongly affected by a temperature variation. By combining experiments and temperature-dependent effective medium theory, we find that this behavior is connected to enhanced damping effects due to electron-phonon scattering. Contrary to standard plasmonic systems, a red-shift of the nonradiative mode occurs for small variations of the environment temperature. Our study shows that, to probe temperature changes, it is essential to exploit nonradiative processes in systems supporting plasmonic excitations, which can be used as very sensitive thermometers via linear absorption spectroscopy.