Are there any non-mathematical abstract objects?

Yes. Anything of which there can be instances is a non-spatiotemporal object, e.g. any property and even any musical work. A musical work is something of which any given performance is a mere instance and with which no performance (or sound-token of any kind) is identical. Also, truths are abstract objects. The fact that I am typing is a redistribution of mass-energy, but not the corresponding truth, which is not. Obviously negative truths, conditional truths, and counterfactual truths cannot possibly be identical with spatiotemporal entities and must therefore be non-spatiotemporal. Finally, mathematical objects, e.g. numbers and functions, at least arguably presuppose the existence of more generic non-spatiotemporal objects, e.g. sets and properties, that are not strictly mathematical in nature.