Anatopsis by Chris Abouzeid
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From the book jacket...

On Ana's thirteenth birthday, the Queen brings a new tutor to the castle: a chilling, corpse-like demigod named Mr. Pound. Mr. Pound is supposed to mold Ana to her mother's wishes, but Ana soon discovers that his plans stretch far beyond the education of a princess. He is searching for the mysterious and powerful Os Divinitas -- and when he finds it, no one will be spared.
Catapulted by an ominous prophecy and a shocking discovery, the Princess and her friends -- including the endearing (if bumbling) Prince Barnaby Georges -- set out on an unlikely quest through the rich world of Anatopsis, where ancient gods, magical Immortals, a rebel army, and the very last dog in the Universe must fight to save their dying worlds.