Thermal transport in aqueous suspensions of Au-core polymer-shell nanoparticles is investigated by time-resolved measurements of optical absorption. The addition of an organic cosolvent to the suspension causes the polystyrene component of the polymer shell to swell, and this change in the microstructure of the shell increases the effective thermal conductivity of the shell by a factor of approximately 2. The corresponding time scale for the cooling of the nanoparticle decreases from 200 ps to approximately 100 ps. The threshold concentration of cosolvent that creates the changes in thermal conductivity, 5 vol % tetrahydrofuran in water or 40 vol % N,N-dimethylformamide in water, is identical to the threshold concentrations for producing small shirts in the frequency of the plasmon resonance. Because the maximum fraction of solvent in the polymer shell is less than 20 vol %, the increase in the effective thermal conductivity of the shell cannot be easily explained by contributions to heat transport by the solvent or enhanced alignment of the polystyrene backbone along the radial direction.