Looks like the Ryzen 7 5800x does NOT come with a stock cooler.
So I will need a cooler when I put it together?
1. You absolutely will need a cooler of some kind. DO NOT POWER ON A SYSTEM WITHOUT A PROPERLY INSTALLED CPU COOLER. Most probably the system will refuse to start, or will shutdown in a few seconds as the thermal protections trip, but if unlucky it may still cause permanent damage.
2. A 5800X is of the more power-efficient end of CPUs for its given number of cores, of around 100-110W under high load. This is nothing compared to high end Intel products; a 12700K for example can draw in excess of 220W.
So if your
case has any proper airflow at all (because the case airflow will determine the surroundings of the CPU and GPU coolers), you can avoid thermal throttling (the CPU automatically reducing capability and power draw as it is nearing maximum temperatures) with practically any aftermarket air cooler. Something like the Deep Cool you mentioned will be more than sufficient. Although I would personally avoid the AMD Wraith options: They're sufficient, but since they're made to be cheap they're little else, and can be quite loud given their small fans.
Since you will need to buy a separate cooler anyway, invest 10-30 dollars more to get a larger
and quieter air cooler (since the more excess capacity it has, the less work the fan will need to do to maintain a steady temperature under load).
A double-fan solution is mostly overkill, and something like the Scythe Fuma 2 will give you benefit only if you plan to go for an ultra-quiet build in all respects (and even then, maybe consider AIO Watercooling solutions if available at similar price-point), or plan to do competitive levels of overclocking on your CPU. That beast is necessary only for high-end Intel products.
Look for a tower cooler with ~4 heatpipes, and one well-sized fan (typically 120-140mm). That will be sufficient. Anything above is extra which can be nice to have, but will not impact performance.
