"Freezing over" is definitely natural in various situations. "The lake froze over last night." "The wheat field is freezing over." I think you might also want to know the phrase "hard freeze", which is a noun. For example, if the weather at night drops far below the freezing point of water, we might say that tonight we will have a "hard freeze". It means there is a strong possibility of all the water freezing solid quickly. If the temperature were below freezing but not quite that far below freezing, the water still might not freeze because it takes time for the water to reduce that far in temperature. But a hard freeze causes the water to freeze quickly. I've never heard anyone say "freeze up".

Hey, these are both good phrasal verbs to describe something that is frozen. Freezing over is used for things such as lakes/water. So I think freezing up would work better for other subjects. For example- The town is really freezing up. I hope this helps :)

What you mean by your question ‘freezing so hard’ is unclear. Freezing by definition is transforming something from a liquid to a solid (hard). Freezing up and melting down are usually used figuratively, not to describe actual freezing. ‘Freezing over’ refers to water where the surface freezes first. If you want to describe very cold temperatures causing things to freeze very quickly, we’d use words like those. I froze up when the teacher called on me unexpectedly. We start playing hockey after the pond freezes over. The ground will freeze solid tonight. The temperature is expected to drop 20 degrees. My freezer’s set to its coldest level. The ice pack will freeze quickly in it.