Learn German: Der Kampfsport kostet ihn viel Zeit. - The martial art costs him a lot of time.

This is the main trick with the verb kosten. In German, kosten is one of a few special verbs that take a double accusative. Instead of having a dative indirect object for the person, it takes an accusative direct object for the person (ihn) and another accusative direct object for the thing it costs (viel Zeit).

So is "viel Zeit" also in the accusative case here?

Exactly! Because kosten forces a double accusative, both the person and the thing are in the accusative case. You can't easily see the case on viel Zeit because die Zeit is a feminine noun, and feminine nouns look exactly the same in the accusative as they do in the nominative.

What makes up the word "Kampfsport"?

It is a compound noun built from der Kampf (the fight or struggle) and der Sport (the sport). Put together, it translates to "martial art" or "combat sport". Remember that the gender of a German compound noun always matches its final word, which is why it is der Kampfsport.

Why isn't there an ending on "viel"? Shouldn't it be "viele Zeit"?