Transparent oxide materials, such as CuAlO2, a p--type transparent conducting oxide (TCO), have recently been studied for high temperature thermoelectric power generators and coolers for waste heat. TCO materials are generally low cost and non--toxic. The potential to engineer them through strain and nano--structuring are two promising avenues toward continuously tuning the electronic and thermal properties to achieve high zT values and low $cost/kW--hr devices. In this work, the strain--dependent lattice thermal conductivity of 2H CuAlO2 is computed by solving the phonon Boltzmann transport equation with interatomic force constants extracted from first--principles calculations. While the average bulk thermal conductivity is around 32 W/(K--m) at room temperature, it drops to between 5--15 W/(K--m) for typical experimental grain sizes from 3nm to 30nm at room temperature. We find that strain can offer both an increase as well as a decrease in the thermal conductivity as expected, however the overall inclusion of small grain sizes dictates the potential for low thermal conductivity in this material.