Thermopower is governed by entropy per carrier, not specific heat, and apparent specific heat scaling appears only for power-law specific heat.
Sugihara, Journal of Physics and Chemistry of Solids 33, 1365 (1972)
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What Really Drives Thermopower: Specific Heat or Entropy as the Unifying Principle Across Magnetic, Superconducting, and Nanoscale Systems
Thermopower is governed by entropy per carrier, not specific heat, and apparent specific heat scaling appears only for power-law specific heat.