For semi-insulating GaAs is the bandgap bigger or smaller than the pure GaAs? The  carrier concentrations  are lower in semi insulating semiconductors, therefore, can we say that their bang gap is wider in terms of energy (eV)?
If they have wider band gap how much bigger is it approximately and what parameters affect it? i.e. is temperature one of them? 
 A: Do you understand what "semi-insulating gallium arsenide" means? I wrote a description recently for wikipedia:

If a GaAs boule is grown with excess arsenic present, it gets certain defects, in particular arsenic antisite defects (an arsenic atom at a gallium atom site within the crystal lattice). The electronic properties of these defects (interacting with others) cause the center of the bandgap to be pinned to the Fermi level, so that this GaAs crystal has very low concentration of electrons and holes. This low carrier concentration is similar to an intrinsic (perfectly undoped) crystal, but much easier to achieve in practice. These crystals are called "semi-insulating", reflecting their high resistivity of 107-109 Ω-cm (which is quite high for a semiconductor, but still much lower than a true insulator like glass).

"Semi-insulating GaAs" is more-or-less synonymous with "GaAs with very few electrons or holes". It has very few electrons and holes because it has very few (effective) dopant atoms, not because it has a bigger bandgap than other GaAs. Semi-insulating GaAs is still GaAs.
Normally, the bandgap of a semiconductor is a fixed material property. GaAs always has 1.4eV bandgap, silicon always has 1.1eV bandgap, etc. etc. Well, the bandgap changes a bit as you increase the electron or hole concentration, but that is usually a small effect until you get to extremely high concentration.
Yes, changing the temperature also changes the bandgap a bit.
