The carbon diffusivity in tungsten is one fundamental and essential factor in the application of tungsten as plasma-facing materials for fusion reactors and substrates for diamond growth. However, data on this are quite scarce and largely scattered. We perform a series of first-principles calculations to predict the diffusion parameters of carbon in tungsten, and evaluate the effect of temperature on them by introducing lattice expansion and phonon vibration. The carbon atom prefers to occupy octahedral interstitial site rather than tetrahedral interstitial site, and the minimum energy path for its diffusion goes through a tetrahedral site. The temperature has little effect on the pre-exponential factor but a marked effect on the activation energy, which linearly increases with the temperature. Our predicted results are well consistent with the experimental data obtained at high temperature (>1800 K) but significantly larger than the experimental results at low temperature (<1800 K).