The Linear Response Theory (LRT) is a widely accepted framework to analyze the power absorption of magnetic nanoparticles for magnetic fluid hyperthermia. Its validity is restricted to low applied fields and/or to highly anisotropic magnetic nanoparticles. Here, we present a systematic experimental analysis and numerical calculations of the specific power absorption for highly anisotropic cobalt ferrite (CoFe2O4) magnetic nanoparticles with different average sizes and in different viscous media. The predominance of Brownian relaxation as the origin of the magnetic losses in these particles is established, and the changes of the Specific Power Absorption (SPA) with the viscosity of the carrier liquid are consistent with the LRT approximation. The impact of viscosity on SPA is relevant for the design of MNPs to heat the intracellular medium during in vitro and in vivo experiments. The combined numerical and experimental analyses presented here shed light on the underlying mechanisms that make highly anisotropic MNPs unsuitable for magnetic hyperthermia.