diff --git a/.github/workflows/publish-package.yml b/.github/workflows/publish-package.yml
index 1b72187f2d..187391b1a3 100644
--- a/.github/workflows/publish-package.yml
+++ b/.github/workflows/publish-package.yml
@@ -11,6 +11,9 @@ on:
     - release/v*
   release:
     types: [published]
+  # Run weekly at 1:23 UTC
+  schedule:
+  - cron:  '23 1 * * 0'
   workflow_dispatch:
     inputs:
       publish:
@@ -44,10 +47,16 @@ jobs:
         python -m pip install build twine
         python -m pip list
 
-    - name: Build a wheel and a sdist
+    - name: Build a sdist and wheel
+      if: github.event_name != 'schedule'
       run: |
         python -m build .
 
+    - name: Build a sdist and wheel and check for warnings
+      if: github.event_name == 'schedule'
+      run: |
+        PYTHONWARNINGS=error,default::DeprecationWarning python -m build .
+
     - name: Verify untagged commits have dev versions
       if: "!startsWith(github.ref, 'refs/tags/')"
       run: |