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

Yes

The word "apesar" is identical in both Brazilian and European Portuguese in terms of meaning, spelling, and grammatical function. In both variants, it is almost exclusively used as part of the prepositional phrase "apesar de" (meaning "despite" or "in spite of"). The only difference between the two variants is pronunciation. In Brazil, the vowels tend to be more open, and the "r" sound varies by region (often being retroflex). In Portugal, vowels are typically more closed or neutralized, and the "r" is more alveolar/tapped.

Brazilian Portuguese Examples

1. Apesar da chuva, nós fomos à praia. (Despite the rain, we went to the beach.)
2. Ela continua sorrindo, apesar de todos os problemas. (She keeps smiling, despite all the problems.)
3. Apesar de ser muito difícil, eu vou tentar. (Despite it being very difficult, I am going to try.)
4. Apesar do cansaço, ele terminou o trabalho. (Despite the tiredness, he finished the work.)
5. Apesar de estar tarde, o bebê ainda está acordado. (Despite it being late, the baby is still awake.)

Portuguese (Continental) Examples

1. Apesar da chuva, fomos à praia. (Despite the rain, we went to the beach.)
2. Ela continua a sorrir, apesar de todos os problemas. (She keeps smiling, despite all the problems.)
3. Apesar de ser muito difícil, vou tentar. (Despite it being very difficult, I will try.)
4. Apesar do cansaço, ele terminou o trabalho. (Despite the tiredness, he finished the work.)
5. Apesar de estar tarde, o bebé ainda está acordado. (Despite it being late, the baby is still awake.)