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This work investigates thermal control of light transmission in a hyperbolic metamaterial based on an array of gold nanorods partially protruding from an anodic aluminum oxide matrix and infiltrated with 5CB liquid crystals. The studied structure exhibits an epsilon-near-zero (ENZ) regime of the effective dielectric permittivity, which appears as a transmission minimum near 680 nm under oblique incidence of p-polarized light. It was experimentally found that heating the structure from 25 to 45 °C leads to a shift of the ENZ-related transmission minimum toward the blue spectral region by about 10 nm and to a fourfold increase in the transmission coefficient. The strongest thermal modulation is observed in the spectral region near the ENZ wavelength. Finite-difference time-domain simulations confirm the observed behavior and show that it is governed by the temperature dependence of the refractive indices of the 5CB liquid crystals. These results demonstrate the potential of hybrid plasmonic metamaterials for thermally tunable photonic devices.