The warm-hot intergalactic medium (warm-hot IGM, or WHIM) pervades the filaments of the Cosmic Web and harbours half of the Universe's baryons. The WHIM's thermodynamic properties are notoriously hard to measure. Here we estimate a galaxy group - WHIM boundary temperature using a new method. In particular, we use a radio image of the giant radio galaxy (giant RG, or GRG) created by NGC 6185, a massive nearby spiral. We analyse this extraordinary object with a Bayesian 3D lobe model and deduce an equipartition pressure $P_\mathrm{eq} = 6 \cdot 10^{-16}\ \mathrm{Pa}$-- among the lowest found in RGs yet. Using an X-ray-based statistical conversion for Fanaroff-Riley II RGs, we find a true lobe pressure $P = 1.5\substack{+1.7\\-0.4}\cdot 10^{-15}\ \mathrm{Pa}$. Cosmic Web reconstructions, group catalogues, and MHD simulations furthermore imply an $\mathrm{Mpc}$-scale IGM density $1 + \delta_\mathrm{IGM} = 40\substack{+30\\-10}$. The buoyantly rising lobes are crushed by the IGM at their inner side, where an approximate balance between IGM and lobe pressure occurs: $P_\mathrm{IGM} \approx P$. The ideal gas law then suggests an IGM temperature $T_\mathrm{IGM} = 11\substack{+12\\-5} \cdot 10^6\ \mathrm{K}$, or $k_\mathrm{B}T_\mathrm{IGM} = 0.9\substack{+1.0\\-0.4}\ \mathrm{keV}$, at the virial radius -- consistent with X-ray-derived temperatures of similarly massive groups. Interestingly, the method is not performing at its limit: in principle, estimates $T_\mathrm{IGM} \sim 4 \cdot 10^6\ \mathrm{K}$ are already possible -- rivalling the lowest X-ray measurements available. The technique's future scope extends from galaxy group outskirts to the WHIM. In conclusion, we demonstrate that observations of GRGs in Cosmic Web filaments are finally sensitive enough to probe the thermodynamics of galaxy groups and beyond.