The azimuthal anisotropy coefficients v2 and v4 of p 0 and. mesons are measured in Au + Au collisions at root s(NN) = 200 GeV as a function of transverse momentum p(T) (1-14 GeV/c) and centrality. The extracted v(2) coefficients are found to be consistent between the two meson species over the measured p(T) range. The ratio of v(4)/v(2)(2) for pi(0) mesons is found to be independent of p(T) for 1- 9 GeV/c, implying a lack of sensitivity of the ratio to the change of underlying physics with p(T). Furthermore, the ratio of v(4)/v(2)(2) is systematically larger in central collisions, which may reflect the combined effects of fluctuations in the initial collision geometry and finite viscosity in the evolving medium.