For most solid materials, the thermal conductivity k decreases with increasing temperature but for AISI 304 stainless steel it increases.
This phenomena of an increasing thermal conductivity with increasing temperature can be observed for many stainless steels (see DOI: 10.1007/978-1-4615-9083-5_12 and 10.1007/978-1-4684-4916-7_20). The reason for this is, that there are two independent terms contributing to the thermal conductivity of metals and their alloys (k = kp + ke), see https://archive.org/details/springer_10.1007-b136496/page/n43/mode/2up . Thereby, each term is affected by the microstructure and composition of the alloy or in this case stainless steel.
The thermal conductivity via the phonons is reduced e.g. by point defects, dislocations or foreign atoms. The phonons are scattered by these defects, which reduces the free path length and thus also the thermal conductivity. Electrons, which also contribute to the thermal conductivity of metals, are scattered by phonons, other electrons, localized magnetic forces, and lattice defects. And each scattering process causes a reduction of the thermal conductivity, this leads to a complex temperature dependent behavior of most steels. Due to the changing conditions (solubility of alloying elements, precipitations, lattice parameters, amount of crystal defects, …), there is a huge variation in the behaviour of the thermal conductivity of steels and stainless steels ( please compare austenitic and martensitic stainless steels). A complex example for tool steels is given in Hafenstein et al (https://doi.org/10.1002/srin.201400597)
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