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

Yes

The spelling, meaning, and grammatical function of "frequentemente" are identical in both Brazilian and European Portuguese. The primary difference lies in pronunciation:

• In Brazil: The "te" syllable is typically palatalized, sounding like "chee" (/tʃi/). The unstressed vowels are generally more open and clearly articulated.
• In Portugal: The "te" syllable is more dental and sharp. Additionally, Portuguese speakers tend to reduce or "swallow" unstressed vowels, making the word sound more clipped and less rhythmic than in Brazil.

Brazilian Portuguese Examples

1. Eu vou frequentemente à academia. (I frequently go to the gym.)
2. Ele viaja frequentemente para o exterior. (He frequently travels abroad.)
3. Nós comemos fora frequentemente. (We eat out frequently.)
4. Ela lê livros frequentemente antes de dormir. (She frequently reads books before sleeping.)
5. O ônibus passa frequentemente por aqui. (The bus passes by here frequently.)

European Portuguese Examples

1. Eu vou frequentemente ao ginásio. (I frequently go to the gym.)
2. Ele viaja frequentemente para o estrangeiro. (He frequently travels abroad.)
3. Nós jantamos fora frequentemente. (We eat out frequently.)
4. Ela lê frequentemente livros antes de dormir. (She frequently reads books before sleeping.)
5. O autocarro passa frequentemente por aqui. (The bus passes by here frequently.)