Internal cooling channels with pin-fin arrays are an important part of gas turbine blade trailing edge design. Short pin-fins act as turbulators in high aspect ratio channels to increase heat transfer and provide structural support to the trailing edge of the blade. Such pin fins however also result in a high pressure drop owing to chaotic flow phenomenon in these highly turbulent flows. High pressure-drop results in higher compressor work due to increased power consumption to push the coolant through these passages. Hence, optimizing the design of pin fin arrays is key to increasing the efficiency of real gas turbine cycles by handling higher turbine inlet temperature and increasing blade life. Moreover, the design process of such pin fin arrays can be computationally very expensive, since it typically involves high-fidelity CFD simulations. The optimization problem involves maximizing Nusselt number, while keeping the friction factor as a constraint. To address this problem, a computationally efficient approach involving Gaussian Processes (GP) surrogate modeling and constrained Bayesian Optimization (BO) has been carried out for optimizing the thermal performance of the pin fin arrays. The multidimensional search space of design parameters includes pin-fin dimensions and shape of the resulting pin-fins. The optimization problem is solved under computational budget limitations and design constraints. A'drop' like optimal design is obtained which has a lower pressure drop and higher Nu compared to the baseline.