A continuous-wave atom laser formed by the outcoupling of atoms from a trapped Bose-Einstein condensate (BEC) potentially has a range of metrological applications. However, in order for the device to be truly continuous, a mechanism to replenish the atoms in the BEC is required. Here we calculate the temporal coherence properties of a continuously pumped atom laser beam outcoupled from a trapped Bose-Einstein condensate that is replenished from a reservoir at finite temperature. We find that the thermal fluctuations of the condensate can significantly decrease the temporal coherence of the output beam due to atomic interactions between the trapped BEC and the beam, and this can impact the metrological usefulness of the device. We demonstrate that a Raman outcoupling scheme imparting a sufficient momentum kick to the atom laser beam can lead to a significantly reduced linewidth.