We perform direct thermovoltage measurements in a single-electron transistor, using on-chip local thermometers, in both the linear and nonlinear regimes. Using a model which accounts for cotunneling, we find excellent agreement with the experimental data with no free parameters even when the temperature difference is larger than the average temperature (far-from-linear regime). This allows us to confirm the sensitivity of the thermovoltage on cotunneling and to find that in the nonlinear regime the temperature of the metallic island is a crucial parameter. Surprisingly, the metallic island tends to overheat even at zero net charge current, resulting in a reduction of the thermovoltage.