If you are making a comparison (which includes superlatives), "in" refers to the population from which you are making the comparison. The biggest X in Y refers to the biggest X of the examples found in Y.
"Across" refers to dispersion; M-W defines it as meaning "throughout". For example, French-speaking people "across" Europe refers to French-speaking people wherever they might be throughout Europe.
So to your examples, you might say, "This food is the most expensive in the world" (correct), or "This food has become expensive across Asia", meaning throughout Asia. When you talk about "across the world", the term "worldwide" is more typically used.
"Japan has the biggest aquarium in the world" would be at least grammatically correct. "Aquariums have become popular across the world" or more idiomatic, "Aquariums have become popular worldwide" would be an example of usage for "across".