  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.
