Personally I run better when it's 40-50°F. The first 400m-800m is kind of weird so I think for an 800m it might not help much, but for 1500m it helps me a bit and for anything between 3k and 21k it helps me a lot.
It seems like the problem with cold for short distances less than 800m is that you have to speed up faster so you're more likely to pull a muscle in the cold, but in longer races since you're not really sprinting immediately it's safer. As long as you warm up properly it shouldn't be too much of a problem.
Also, the clothes you wear in the cold can also affect your race, like if you wear sweatpants your time will obviously be a bit different than if you wore shorts. Where I live people usually race in long sleeves if it's under 50°F but most will still wear shorts unless it's below freezing (the climate here is a bit colder than in other places though so we might just be more used to it).