  Recently, Heusler ferromagnets have been found to exhibit unconventional
anomalous electric, thermal, and thermoelectric transport properties. In this
study, we employed first-principles density functional theory calculations to
systematically investigate both intrinsic and extrinsic contributions to the
anomalous Hall effect (AHE), anomalous Nernst effect (ANE), and anomalous
thermal Hall effect (ATHE) in two Heusler ferromagnets: Fe$_2$CoAl and
Fe$_2$NiAl. Our analysis reveals that the extrinsic mechanism originating from
disorder dominates the AHE and ATHE in Fe$_2$CoAl , primarily due to the steep
band dispersions across the Fermi energy and corresponding high longitudinal
electronic conductivity. Conversely, the intrinsic Berry phase mechanism,
physically linked to nearly flat bands around the Fermi energy and gapped by
spin-orbit interaction band crossings, governs the AHE and ATHE in Fe$_2$NiAl.
With respect to ANE, both intrinsic and extrinsic mechanisms are competing in
Fe$_2$CoAl as well as in Fe$_2$NiAl. Furthermore, Fe$_2$CoAl and Fe$_2$NiAl
exhibit tunable and remarkably pronounced anomalous transport properties. For
instance, the anomalous Nernst and anomalous thermal Hall conductivities in
Fe$_2$NiAl attain giant values of 8.29 A/Km and 1.19 W/Km, respectively, at
room temperature. To provide a useful comparison, we also thoroughly
investigated the anomalous transport properties of Co$_2$MnGa. Our findings
suggest that Heusler ferromagnets Fe$_2$CoAl and Fe$_2$NiAl are promising
candidates for spintronics and spin-caloritronics applications.
