This depends on what you mean by "cold" and there are several other factors as well.

For one, they work great below 28 degrees. For the area between 28-34, there can be issues. In this range, water will more readily condense out of the air and form ice on the outdoor unit. And if it is raining and it is 34 out, you'll really have some ice.

But below 28 degrees... any water in the air is already "frozen" and you aren't going to have as big of an issue of ice spontaneously forming on a colder surface.

As long as you can get air flow across the coils on the outdoor unit you are fine, in one sense, "the colder, the better".

But that leads to the next issue: what to do when you do have ice blocking air flow? And this is when price comes into play. To my knowledge, all of the lower tier brand names (goodman, payne, bryant, maybe even ruud and rheem) will use a timer based defrost control. Basically, once the outdoor coil goes below 32 degrees, a switch is tripped and a defrost cycle will be forced after 60 minutes, whether there is ice on the unit or not. When temps are below freezing, a defrost cycle could easily take 20 minutes of extra runtime to recover the temperature. Even worse, if snow is drifted up against outdoor unit, a defrost cycle will cause it to melt into the coils which will turn into ice once the defrost cycle is over. So an unnecessary defrost could take a completely ice free outdoor unit and leave it with one side caked in ice.

To combat this issue, most of your top line brands (Trane, American Standard, Lennox, Carrier) will have "on demand" defrost so you might very well go 4+ hours of runtime and never see a defrost. However, each brand has their own quirks and can still end up with unnecessary defrost cycles if the air flow through the indoor unit (dirty filter) is poor or if the system refrigeration charge is not 100% perfect.

The other thing that seems to get people is run times. In the south in 100+ degree temps, you can expect your A/C to run for 12 hours in a 24 hour period. Yet for some reason, when a heatpump runs for 2+ hours straight when it is 20 degrees outside, people flip out that it's running too long and going to blow your electric bill up so they flip it over to electric only/emergency mode.

Let's do the math... your heat pump is running for hours on end, drawing 3kwh every hour. You freak out and flip it to emergency mode which turns on a 20kw electric heater and the unit now runs for 30 minutes followed by a 30 minute off cycle. You think it is only using "half" the electricity because it is running half as much. But the reality is, you are now using 10kwh every hour.

Call me crazy, but I don't think it was politics or frozen natural gas that lead to the 2021 Texas blizzard power outage, but people with heat pumps that have no idea what they are doing. Even before temps were freezing, several local community groups on facebook had people spamming "it's going to get below 32 tonight, so for you heat pump users, make sure you put it in emergency heat mode!". And then to make matters worse, local HVAC companies, with the large influx of people complaining their heat pump had been running for "hours on end", start chiming in saying "it's too cold, go to emergency mode"... Meanwhile, I'm somewhat new to heat pumps myself, but I had forced mine to only use electric heat when in defrost mode. It ran flawlessly, I was very impressed. The vents were blowing 90 degrees all the way until it was 20 degrees out. Once it was 6 degrees out, it was blowing 81 out the vents, but still enough to hold temperature in the house (66). On the worst day, I had a combined runtime of 16 hours with 7 defrost cycles. My bill for the whole month was $140 (the highest ever), while all my neighbors that tried to "save" money by going to emergency mode had bills of $300+.