We use the tight-binding model to study the effect of transverse electric field on the low-temperature electronic specific heat (Cv) for armchair and zigzag boron nitride nanotubes (ABNNTs and ZBNNTs). For wide-band-gap BNNTs, electric field could significantly modulate their energy dispersions and shift many electronic states close to the Fermi energy. Under a critical electric field (Fc) the density of states show special peak structures and the vanishing specific heat at zero field jumps to a giant one. Cv, at Fc's, has a value comparable to that of the phonon specific heat and reveals strongly non-linear dependence on temperature. The critical field strength and the value of giant specific heat are closely related to nanotube's geometry. In the presence of Fc's, the extra longitudinal magnetic flux could enhance the value of Cv again at low temperature for ZBNNTs, whereas it is not always true for ABNNTs.