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

Yes

The word "recentemente" is identical in meaning, spelling, grammar, and typical usage in both Brazilian and Continental Portuguese. The only difference lies in pronunciation.

In Brazilian Portuguese, the unstressed vowels are generally more open and clearly articulated. In Continental Portuguese, unstressed vowels (especially the "e" sounds in this word) tend to be much more reduced, "shorter," or even almost silent, leading to a more "closed" phonetic rhythm.

Brazilian Portuguese Examples

1. Eu li um livro muito bom recentemente. (I read a very good book recently.)
2. Ela mudou de emprego recentemente. (She changed jobs recently.)
3. Não tenho visto ele recentemente. (I haven't seen him recently.)
4. Recentemente, comecei a fazer academia. (Recently, I started going to the gym.)
5. Tudo mudou muito recentemente por aqui. (Everything changed a lot around here recently.)

Continental Portuguese Examples

1. Tenho lido um livro muito bom recentemente. (I have been reading a very good book recently.)
2. Ela mudou de emprego recentemente. (She changed jobs recently.)
3. Não o tenho visto recentemente. (I haven't seen him recently.)
4. Recentemente, comecei a frequentar o ginásio. (Recently, I started going to the gym.)
5. Tudo mudou muito recentemente por este lado. (Everything changed a lot around here recently.)