The question in the title is a real question asked by people, but it is actually misleading. The story about the EU capital is a bit more complicated.
If you take the question “What is the most boring big capital city in Europe?” and use it on Google, you will see that there are several forums and other sites, all pointing out to the same city.
There are cities that developed historically as separate cities but in time they became one city. Here you can see such examples regarding capital cities of some countries.
When this is about the hottest capital city in the world, there are several contenders. The differences in their average yearly temperatures are so small that it is difficult to say which is the warmest.
When this is about the smallest capitals in the world, you will usually see Vatican mentioned. This is a famous city-state with a unique position and history, but it is not the smallest.
There are five US capitals with names from Native American languages, and there are also tens of non-capital cities with the same name origin.
It may be difficult to provide exact information about the coldest capital city in the world because this depends on reliability of the available data. The temperature differences among the coldest cities are quite small.
Strictly speaking, there are only two states in the USA that share the name with their capitals. These are Oklahoma and Indiana.
So did you know that some countries decide to split capital functions and have them seated in two separate cities? Various reasons are behind such moves, dependent on the country.
There are currently far over 30 countries around the globe that have changed their capital cities, and some have done this a few times in the past. Here you can read about them and about reasons for this.