Datetime example field result in build fail for Sphinx

I'm trying to use this plugin to integrate an OpenAPI spec in a Sphinx document. However, when using date-time strings in examples I get a build error. An example spec which creates the error would be the following

openapi: 3.0.1
info:
  title: EXAMPLE DATETIME
  version: 1.0.0
paths:
  /test:
    get:
      summary: Test endpoint
      operationId: testId
      responses:
        "200":
          description: Successful operation
          content:
            application/json:
              schema:
                $ref: '#/components/schemas/TestResponse'
components:
  schemas:
    TestResponse:
      type: object
      properties:
        TestDate:
          type: string
          format: date-time
      example:
        TestDate: 2000-01-23T04:56:07.000+00:00

Trying to use this file results in:

File "/home/user/.local/share/hatch/env/virtual/docs/0DJxZvn8/docs/lib/python3.12/site-packages/sphinxcontrib/openapi/directive.py", line 56, in run
    return renderer_cls(self.state, self.options).render(spec)
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/home/user/.local/share/hatch/env/virtual/docs/0DJxZvn8/docs/lib/python3.12/site-packages/sphinxcontrib/openapi/renderers/abc.py", line 40, in render
    for line in self.render_restructuredtext_markup(spec):
  File "/home/user/.local/share/hatch/env/virtual/docs/0DJxZvn8/docs/lib/python3.12/site-packages/sphinxcontrib/openapi/renderers/_httpdomain_old.py", line 55, in render_restructuredtext_markup
    yield from openapihttpdomain(spec, **self._options)
  File "/home/user/.local/share/hatch/env/virtual/docs/0DJxZvn8/docs/lib/python3.12/site-packages/sphinxcontrib/openapi/openapi30.py", line 342, in _httpresource
    for line in _example(
  File "/home/user/.local/share/hatch/env/virtual/docs/0DJxZvn8/docs/lib/python3.12/site-packages/sphinxcontrib/openapi/openapi30.py", line 211, in _example
    example['value'] = json.dumps(
                       ^^^^^^^^^^^
  File "/usr/lib/python3.12/json/__init__.py", line 238, in dumps
    **kw).encode(obj)
          ^^^^^^^^^^^
  File "/usr/lib/python3.12/json/encoder.py", line 202, in encode
    chunks = list(chunks)
             ^^^^^^^^^^^^
  File "/usr/lib/python3.12/json/encoder.py", line 432, in _iterencode
    yield from _iterencode_dict(o, _current_indent_level)
  File "/usr/lib/python3.12/json/encoder.py", line 406, in _iterencode_dict
    yield from chunks
  File "/usr/lib/python3.12/json/encoder.py", line 439, in _iterencode
    o = _default(o)
        ^^^^^^^^^^^
  File "/usr/lib/python3.12/json/encoder.py", line 180, in default
    raise TypeError(f'Object of type {o.__class__.__name__} '
TypeError: Object of type datetime is not JSON serializable

Exception occurred:
  File "/usr/lib/python3.12/json/encoder.py", line 180, in default
    raise TypeError(f'Object of type {o.__class__.__name__} '
TypeError: Object of type datetime is not JSON serializable
The full traceback has been saved in /tmp/sphinx-err-vn0id8op.log, if you want to report the issue to the developers.
Please also report this if it was a user error, so that a better error message can be provided next time.
A bug report can be filed in the tracker at <https://github.com/sphinx-doc/sphinx/issues>. Thanks!

If I replace the TestDate entry in the example with 200001-23T04:56:07.000+00:00 the error doesn't get triggered.

Start with sphinxcontrib/openapi/openapi30.py at the _example function and its json.dumps call, then reproduce the failure with the YAML spec and date-time value from the issue. Check how the example is parsed before serialization. Done means the Sphinx build handles this valid date-time example without the datetime serialization error.