Thermal properties of glueballs in SU(3) Yang-Mills theory are investigated in a large temperature range from 0.3T(c) to 1.9T(c) on anisotropic lattices. The glueball operators are optimized for the projection of the ground states by the variational method with a smearing scheme. Their thermal correlators are calculated in all 20 symmetry channels. It is found in all channels that the pole masses M-G of glueballs remain almost constant when the temperature is approaching the critical temperature T-c from below, and start to reduce gradually with the temperature going above T-c. The correlators in the 0(++), 0(-+), and 2(++) channels are also analyzed based on the Breit-Wigner Ansatz by assuming a thermal width Gamma to the pole mass omega(0) of each thermal glueball ground state. While the values of omega(0) are insensitive to T in the whole temperature range, the thermal widths Gamma exhibit distinct behaviors at temperatures below and above T-c. The widths are very small (approximately a few percent of omega(0) or even smaller) when T < T-c, but grow abruptly when T > T-c and reach values of roughly Gamma similar to omega(0)/2 at T approximate to 1.9T(c).