We propose an L2-norm based test for testing the equality of the mean functions of k groups of weakly dependent stationary functional time series. The proposed testing procedure is flexible and can be applied to both homoscedastic and heteroscedastic cases. Under the null hypothesis, the asymptotic random expression of the test statistic is a χ2-type mixture, which is approximated by a two-cumulant and a three-cumulant matched χ2 approximation methods, respectively. Under a local alternative hypothesis, the asymptotic random expression is also derived and the test is shown to be root-n consistent. Simulation studies are performed to compare the finite sample performance of the proposed test under various scenarios with alternatives e.g. an existing FPCA based test and some respective ANOVA tests. It is shown that the proposed test generally outperforms the alternative tests in terms of empirical sizes and powers. Two real data examples help to illustrate the implementation of our test based on the US yield curves and Google flu trends, respectively.