I would like to point out that the actions of the Governor General of Australia are largely determined by convention, rather than codified laws.
So if Peter Cosgrove wants behave like the spineless lackey the Government appointed him to be, he is allowed to do so. If he decides to recall parliament using an obscure technicality in the constitution which no Governor General has had the hide to use since 1977, he is entitled to do so. If the Governor General wishes to act in a churlish and immature manner by snubbing the deputy leader of the Opposition in retaliation for criticism of his actions to the cheers of Coalition MPs, he is allowed to do so.
If Peter Cosgrove wants to demean his position as the representative of this nation's head of state because it politically benefits his mates, there is nothing in the Australian Constitution which prevents him from doing so.
The actions of the Governor General are shaped by convention, not bound by them. There is nothing in the constitution which specifically states that the Governor General must act in a mature, dignified and impartial manner.
So let the man get on with his $425,000 per annum taxpayer funded job. He just doing what the Constitution doesn't specifically say he can't do.