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 h 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/(m K) at room temperature, it drops to between 5 and 15 W/(m K) for typical experimental grain sizes from 3 nm to 30 nm. 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.