Does the bonus boost die from improved inspiring rhetoric stack. For example.if I have a politico perform the action twice in a single turn can i give my party two boost or just one?

This is a guess, but... in the description, it says "Each ally affected by Inspiring Rhetoric gains [boost] on all skill checks..." If you look at the errata for Cover, the entry was changed from "increases the character's ranged defense by 1" to "allows the character to gain ranged defense 1", meaning that cover does not give you an additional ranged defense, but just gives you Ranged Defense 1. So, if you already have Ranged Defense of 1 or higher, cover does nothing.

I think we can safely say that Improved Inspiring Rhetoric works the same way, in that the characters gain the boost die, rather than adding a boost die to the skill roll. So, I would say that it doesn't stack.

Again, just my take

FFG seems try to be consistant when they say "gain" or "increase". If it says "gain", as OggDude said, then I'd agree with him that you can't use Improved Inspiring Rhetoric more than once to stack bonuses.

Also, the bonus stacking could get crazy if it was intended. Doesn't it last for a number of rounds per rank in the skill? You can also get talents to do it as a Maneuver. So, you could do it three times in a turn (two maneuvers and an action) to grant three boost dice. If you have 5 ranks on the skill, then you'd be granting 15 boost die on the fifth turn you did this. That alone would make me want to not allow boost dice to stack, even if it was possible by the rules.

Does the bonus boost die from improved inspiring rhetoric stack. For example.if I have a politico perform the action twice in a single turn can i give my party two boost or just one?

My book is in the other room and I'm far too lazy to get them, but I seem to recall the text saying that yes the bonuses (and minuses for Scathing Tirade) don't stack. Roll as much as you like, you only get one black (or blue).

The only benefit to multiple uses of Inspiring Rhetoric is the ability to rapidly reduce the Strain your group is suffering. This is pretty significant.