fix: add link check tox env

tox.ini

@@ -4,7 +4,7 @@ envlist =
     code-style
     doc-style
     {py38,py39,py310,py311,py312}{,-coverage}
-    doc
+    doc-{links,html}
 skip_missing_interpreters = true
 isolated_build = true
 isolated_build_env = build
@@ -43,8 +43,14 @@ commands =
     vale sync --config="{toxinidir}/doc/.vale.ini"
     vale --config="{toxinidir}/doc/.vale.ini" "{toxinidir}/doc" 
 
-[testenv:doc]
+[testenv:doc-{links, html}]
 description = Check if documentation generates properly
+setenv =
+    links: BUILDER = linkcheck
+    html: BUILDER = html
+passenv =
+    BUILD_API
+    BUILD_EXAMPLES
 extras = doc
 commands =
-    sphinx-build -d "{toxworkdir}/doc_doctree" doc/source "{toxworkdir}/doc_out" --color -vW -bhtml
+    sphinx-build -d "{toxworkdir}/doc_doctree" doc/source "{toxinidir}/doc/_build/{env:BUILDER}" --color -vW -b {env:BUILDER} -j auto