Based on the Brayton cycle for gas-turbine engines, the high thermal efficiency and power output of a gas-turbine engine can be obtainable when the gas-turbine engine operates at high turbine inlet temperatures. However, turbine components e.g., inlet guide vane, rotor blade, and stator vane request high cooling performance. Typically, internal cooling and film cooling are two effective techniques that are widely used to protect high thermal loads for the turbine components in a state-of-the-art gas turbine. Consequently, the high thermal efficiency and power output can be obtained, and the turbine lifespan can be prolonged, also. On top of that, a comprehensive understanding of flow and heat transfer phenomena in the turbine components is very important. As a result, both experiments and simulations have been used to improve the cooling performance of the turbine components. In fact, the cooling air used in the internal cooling and film cooling is partially extracted from the compressor. Therefore, variations in the cooling air affect the cooling performance of the turbine components directly. This paper presents a numerical study on the influence of the cooling air on cooling-performance sensitivity of an internally convective turbine vane, MARK II using the computational fluid dynamics (CFD)/conjugate heat transfer (CHT) with the SST k-ω turbulence model. Result comparisons are conducted in terms of pressure, temperature, and cooling effectiveness under the effects of the inlet temperature, mass flow rate, turbulence intensity, and flow direction of the cooling air. The cooling-performance sensitivity to the coolant parameters is shown through variations of local cooling effectiveness, and area and volume-weighted average cooling effectiveness.