  Co$_3$Sn$_2$S$_2$ is a ferromagnetic semi-metal with Weyl nodes in its band
structure and a large anomalous Hall effect below its Curie temperature of 177
K. We present a detailed study of its Fermi surface and examine the relevance
of the anomalous transverse Wiedemann Franz law to it. We studied Shubnikov-de
Haas oscillations along two orientations in single crystals with a mobility as
high as $2.7\times$10$^3$ cm$^2$V$^{-1}$s$^{-1}$ subject to a magnetic field as
large as $\sim$ 60 T. The angle dependence of the frequencies is in agreement
with density functional theory (DFT) calculations and reveals two types of hole
pockets (H1, H2) and two types of electron pockets (E1, E2). An additional
unexpected frequency emerges at high magnetic field. We attribute it to
magnetic breakdown between the hole pocket H2 and the electron pocket E2, since
it is close to the sum of the E2 and H2 fundamental frequencies. By measuring
the anomalous thermal and electrical Hall conductivities, we quantified the
anomalous transverse Lorenz ratio, which is close to the Sommerfeld ratio
($L_0=\frac{\pi^2}{3}\frac{k_B^2}{e^2}$) below 100 K and deviates downwards at
higher temperatures. This finite temperature deviation from the anomalous
Wiedemann-Franz law is a source of information on the distance between the
sources and sinks of the Berry curvature and the chemical potential.
