The pysha3 module you found was based on an draft of the SHA-3 specification, before it was standardised.
The module was created as a POC for Python issue 16113, and the code has not been updated since 2012. The NIST standard wasn't finalised until October 2015. As such, the implementation can't be used if you expect it to follow the released standard.
That ticket links to an implementation that does claim to have been updated to the standard: https://github.com/bjornedstrom/python-sha3. That package doesn't appear to be listed on PyPI, but can be installed with pip directly from GitHub:
pip install git+https://github.com/bjornedstrom/python-sha3
and this package does produce the expected result:
>>> import hashlib
>>> import sha3
>>> hashlib.sha3_512(b'').hexdigest()
b'a69f73cca23a9ac5c8b567dc185a756e97c982164fe25859e0d1dcc1475c80a615b2123af1f5f94c11e3e9402c3ac558f500199d95b6d3e301758586281dcd26'
This package doesn't patch the built-in hashlib.new()
constructor, but that's easily done by plugging in the constructor into the module cache:
>>> hashlib.__builtin_constructor_cache['sha3_512'] = sha3.sha3_512
>>> hashlib.new('sha3_512')
<sha3.SHA3512 object at 0x10b381a90>