I didn't altogether like Gaspode in his earlier appearances (in MP and MAA) but I came to like him (and perhaps understand Pratchett's use of him) a good deal more in this book.
Gaspode is, in some ways, the canine version of Angua. That is, he has some human attributes (speech and increased intelligence over the ordinary dog), but in other ways he wants to be a "good dog" and have a benevolent master--only he doesn't want that either when Carrot, rather than providing for him himself in MAA, asks Vetinari to find a home for him.
Gaspode has in some ways a major role in this book, and because of what he learns about humans (especially Carrot), when he returns to Ankh-Morpork on his own, he joins the Canting Crew and becomes invisible. I suspect that Pratchett, who had painted himself into a corner in his development of Carrot in previous books, tried to use this one to make him somewhat more human and less "divine kingly". And part of this effort is to give Carrot a dark side, and in that Gaspode plays a major part.
Carrot unashamedly manipulates Gaspode into helping him track Angua and promises steak every night. (Gaspode gets the odd chicken, but no steak ever.) Gaspode, as he has ever since MP, tends to see himself in "movie role" terms (Brave Dog Defends Master from Wolves). But throughout, Gaspode is the only reason Carrot can track Angua at all. Carrot, who has come off totally unprepared, nearly gets both of them killed before they are rescued by Angua and Gavin. Gaspode does his best to save Carrot's life from attack by what he thinks are "ravening wolves." But Carrot, rather than acknowledging Gaspode's heroic actions, explains his presence to Vimes as "Yes, he . . .helped me get here." Carrot, of course, would have failed miserably in his chase without Gaspode.
And Gaspode is, of course, instrumental in the fight between Wolfgang and Gavin, and goes (along with the others) into the river. And intelligent thought by Angua would have indicated that there was good reason to believe that the howl was started by Gaspode (as indeed it was). But neither Angua nor Carrot make any effort to look for him when they go to bury Gavin. Carrot, when told about Gaspode's apparent fate by Sam, makes his usual banal comment, "Poor little soul. He was a good dog at heart."
Pratchett tried to redeem Carrot (unsuccessfully) by telling us the words which would have sounded trite and wrong on anyone else are somehow redeemed by the fact it is Carrot saying them. They sound trite and banal anyhow, and Pratchett's statement doesn't change anything. Rather it makes Carrot sound like a pompous ass. But this is an indication of Pratchett's failure to redeem Carrot--to make him a viable, useful character.
Carrot, who once was "a man of the city" henceforth is only concerned with his own importance, pleasures and self-image. Significantly Carrot has extremely small roles to play in subsequent books.
Gaspode, on the other hand, has a major role in The Truth where he now steers absolutely clear of all humans other than the canting crew who serve as camouflage for his remarkable skills and talents. No wonder he doesn't trust "sane" normal humans any further after his experiences in 5th Elephant.