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Considering the growing use of cellulose in various applications, knowledge and understanding of its physical properties become increasingly important. Thermal conductivity is a key property, but its variation with porosity and density is unknown, and it is not known if such a variation is affected by fiber size and temperature. Here, we determine the relationships by measurements of the thermal conductivity of cellulose fibers (CFs) and cellulose nanofibers (CNFs) derived from commercial birch pulp as a function of pressure and temperature. The results show that the thermal conductivity varies relatively weakly with density (ρ_sample = 1340–1560 kg m^–3) and that its temperature dependence is independent of density, porosity, and fiber size for temperatures in the range 80–380 K. The universal temperature and density dependencies of the thermal conductivity of a random network of CNFs are described by a third-order polynomial function (SI-units): κ_CNF = (0.0787 + 2.73 × 10^–3·T – 7.6749 × 10^–6·T² + 8.4637 × 10^–9·T³)·(ρ_sample/ρ₀)², where ρ₀ = 1340 kg m^–3 and κ_CF = 1.065·κ_CNF. Despite a relatively high degree of crystallinity, both CF and CNF samples show amorphous-like thermal conductivity, that is, it increases with increasing temperature. This appears to be due to the nano-sized elementary fibrils of cellulose, which explains that the thermal conductivity of CNFs and CFs shows identical behavior and differs by only ca. 6%. The nano-sized fibrils effectively limit the phonon mean free path to a few nanometers for heat conduction across fibers, and it is only significantly longer for highly directed heat conduction along fibers. This feature of cellulose makes it easier to apply in applications that require low thermal conductivity combined with high strength; the weak density dependence of the thermal conductivity is a particularly useful property when the material is subjected to high loads. The results for thermal conductivity also suggest that the crystalline structures of cellulose remain stable up to at least 0.7 GPa.