Asian rainbow options provide investors with a new option solution as an effective tool for asset allocation and risk management. In this paper, we address the pricing problem of Asian rainbow options with stochastic interest rates that obey the Vasicek model. By introducing the Vasicek model as the change process of the stochastic interest rate, based on the non-arbitrage principle and the stochastic differential equation, the number of assets of the Asian rainbow option is expanded to $n$ dimensions, and the pricing formulas of the Asian rainbow option with multiple ( $n$) assets under the Vasicek interest rate model are obtained. The multi-asset pricing results under stochastic interest rates provide more possibilities for Asian rainbow options. Furthermore, Monte Carlo simulation experiments show that the pricing formula is accurate and efficient under double stochastic errors. Finally, we perform parameter sensitivity analysis to further justify the pricing model.