We are investigating the output and temperature characteristics of Yb:YAG TRAM (Total-Reflection Active Mirror) laser using zero-phonon line excitation (969-nm pumping) and direct water jet cooling for efficient heat removal. The TRAM configuration has an advantage of cooling the surface of the Yb:YAG disk without the high-reflection coating. We have developed an efficient hydrodynamic cooling system, where the disk is directly cooled by impinging water jet with flow rate of up to 52 liter/min., while the water temperature can be controlled from 7 to 80 degrees Celsius. For the estimation of operating temperatures of the Yb:YAG, we measured fluorescence spectra from Yb:YAG using a spectrometer. We tested several types of TRAM with different layer thicknesses and doping concentrations, which were designed to absorb more than 80% of the pump power in a single bounce at room temperature. A fiber-coupled CW laser diode (FCLD) with 600 W output power at 969 nm was used as a pump source. The dependences of oscillator output power and the laser medium temperature on the cooling water temperature and flow rate were investigated. The direct impinging water jet at high flow rate was demonstrated to be effective for cooling the laser medium. It was also confirmed that the zero-phonon line excitation at 969-nm resulted in lower laser medium temperature and hence higher output power compared to the 940-nm pumping. In addition, we demonstrated kW-class laser oscillation using the cooling system and achieved slope efficiency of 63 %.