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

No.

While the spelling and core meanings of "gasto" are identical in both variants, there are differences in pronunciation and typical usage. In European Portuguese, unstressed vowels are more closed or reduced (the final "o" sounds more like a short "u" or a neutral "e"), whereas in Brazilian Portuguese, they are more open. Regarding usage, Brazilians frequently use "gasto" colloquially to mean "exhausted" when referring to people; a Portuguese person would rarely use it this way, preferring "cansado" or "exausto." Additionally, while both use "gasto" for expenses, Portuguese speakers more frequently use the word "despesa" for recurring costs or bills.

Brazilian Portuguese Examples

1. O meu gasto com comida aumentou muito. (My spending on food increased a lot.)
2. Este tênis já está bem gasto. (This sneaker is already quite worn out.)
3. Estou muito gasto depois do trabalho. (I am very exhausted after work.)
4. Não quero ter um gasto de tempo desnecessário. (I don't want to have an unnecessary waste of time.)
5. O gasto com a conta de luz foi alto. (The expense with the electricity bill was high.)

European Portuguese Examples

1. A minha despesa com alimentação aumentou muito. (My expenditure on food increased a lot.)
2. Este sapato já está bem gasto. (This shoe is already quite worn out.)
3. Estou muito cansado depois do trabalho. (I am very tired after work.)
4. Não quero perder tempo desnecessariamente. (I don't want to waste time unnecessarily.)
5. O valor da fatura da eletricidade foi alto. (The amount of the electricity bill was high.)