Based on the Sommerfeld diffraction integral, the anomalous spectral properties of an ultrashort pulsed Hermite-Gaussian beams propagating in dispersive media are studied , and its comparisons with anomalous spectral behaviors in free space are analyzed. It is found that the on-axis spectrum blueshift of ultrashort pulsed Hermite-Gaussian beams is relative to group velocity and pulse duration, but not to group velocity dispersion. The size of relative spectral shift in dispersive media is slightly smaller than that in free space, and the relative spectral shifts decrease with increasing the pulse duration. The location of spectral switching is relative to the transversal distance, transmission media,and the Hermite function indexes m and n, but not to pulse duration. The transversal distance of spectral switching in dispersive media is less than that in free space, the relative spectral shift slowly decreases as the order number of the spectral switching increases.