The semiparametric accelerated failure time (AFT) model is a popular linear model in survival analysis. AFT model and its associated inference methods assume homoscedasticity of the survival data. It is shown that violation of this assumption will lead to inefficient parameter estimation and anti-conservative confidence interval estimation, and thus, misleading conclusions in survival data analysis. However, there is no valid statistical test proposed to test the homoscedasticity assumption. In this paper, we propose the first novel quasi-likelihood ratio test for the homoscedasticity assumption in the AFT model. Simulation studies show the test performs well. A real dataset is used to demonstrate the usefulness of the developed test.