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Reproducible Example

import pandas
df = pandas.DataFrame([['a', 'b'], ['c', 'd']])
path = 'test.csv'
df.to_csv(path)
df = pandas.read_csv(path, dtype='string[pyarrow]', index_col=0)
df.index # Index dtype is string.
df = pandas.read_csv(path, dtype=str, index_col=0)
df.index # Index dtype is int64.

Issue Description

I'm not sure if this is intended or not. If you use the index_col parameter in read_csv with the 'string[pyarrow]' dtype the index dtype is string, but if you use the str dtype or don't specify a dtype it will be int64. It has been my experience with pandas that the dtype only refers to the data and not the index. I initially had a few bugs stemming from switching to the 'string[pyarrow]' dtype and the resulting index type change. I also tested read_excel, and it has an index with an int64 dtype when specifying a 'string[pyarrow]' dtype, so there is also inconsistency when reading different table formats.

Expected Behavior

The expected behavior is that the index is the same dtype regardless of the specified dtype in read_csv, or that the index dtype is always affected by the specified dtype, and that this behavior is the same for read_excel.

Installed Versions