Cross-plane electrical and thermal transport in thin films of a conducting polymer (poly(3,4-ethylenedioxythiophene), PEDOT) stabilized with trifluoromethanesulfonate (OTf) is investigated in this study. We explore their electrical properties by conductive atomic force microscopy (C-AFM), which reveals the presence of highly conductive nano-domains. Thermal conductivity in cross-plane direction is measured with Null-Point scanning thermal microscopy (NP-SThM): PEDOT:OTf indeed demonstrates non-negligible electronic contribution to the thermal transport. We further investigate the correlation between electrical and thermal conductivity by applying post-treatment: chemical reduction (de-doping) for the purpose of lowering charge carrier concentration and hence, electrical conductivity and acid treatment (over-doping) to increase the latter. From our measurements, we find a vibrational thermal conductivity of 0.34±0.04 W m-1 K-1. From the linear dependence or the electronic contribution of thermal conductivity vs. the electronic conductivity (Widemann-Franz law), we infer a Lorenz number 6 times larger than the classical Sommerfeld value as also observed in many organic materials for in-plane thermal transport. Applying the recently proposed molecular Widemann-Franz law, we deduced a reorganization energy of 0.53±0.06 eV