Is the word gosto the same in Brazilian and European Portuguese?

Yes

The word "gosto" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. It functions as the first-person singular present indicative of the verb gostar (to like) and as a noun meaning "taste," "flavor," or "preference." The only difference is pronunciation. In Brazilian Portuguese, the vowels are generally more open and the rhythm is more syllable-timed. In Continental Portuguese, unstressed vowels undergo significant reduction; consequently, the "o" at the end of "gosto" is often much more closed, muffled, or even nearly disappears, making the word sound much more compressed.

Brazilian Portuguese Examples

1. Eu gosto muito de chocolate. (I like chocolate a lot.)
2. Você gosta de viajar pelo Brasil? (Do you like to travel through Brazil?)
3. O gosto desse suco está muito bom. (The taste of this juice is very good.)
4. A gente gosta de ir à praia. (We like to go to the beach.)
5. Não gosto de acordar cedo. (I don't like waking up early.)

Continental Portuguese Examples

1. Eu gosto muito de chocolate. (I like chocolate a lot.)
2. Tu gostas de viajar por Portugal? (Do you like to travel through Portugal?)
3. O gosto deste sumo está muito bom. (The taste of this juice is very good.)
4. Nós gostamos de ir à praia. (We like to go to the beach.)
5. Não gosto de acordar cedo. (I don't like waking up early.)