The dog showed no loss of skill from cryogenics. But testing on a dog's tricks isn't the same as cognitive assessment of the full range.
But we didn't damper the dog, as we don't have damper tech. The dog was frozen solid for up to a week. -70°C (~203°K). It had no brain activity during freeze. Therefore, the consciousness of the dog, however much thay may have been, apparently stopped. We don't have any proof that mind is anything more than chemical reactions; while theologians often assert that the mind is the junction of soul and brain, we have no proof of that. Yet. (Especially since the vast majority of theologians don't ascribe souls to dogs, and only very few would do so even for primates.) The dog suffered no personality changes notable to the experimenters, but again, dog's are not good subjects for personality assessment.
Thermodynaimcally, the chemical reactions are nil; ain't no brain activity to form a mind, even if we assume mind = soul + brain activity. So it's on hold.
Now, into handwaivium land, if we stop the reactivity by use of dampers to force molecular motion down, and likewise prevent nuclear decay, the net effect is the same.
When we freeze people by either means, we get no brain activity. The only way we know, in the present, real world, to stop a brain and restart it is to freeze it and then thaw it, after having it metabolize an anti-icing agent.