diff --git a/sphinx/util/osutil.py b/sphinx/util/osutil.py
index 20c6a90cc..5bfbcabc4 100644
--- a/sphinx/util/osutil.py
+++ b/sphinx/util/osutil.py
@@ -151,15 +151,23 @@ no_fn_re = re.compile(r'[^a-zA-Z0-9_-]')
 def make_filename(string):
     return no_fn_re.sub('', string) or 'sphinx'
 
-if PY2:
+
+def ustrftime(format, *args):
     # strftime for unicode strings
-    def ustrftime(format, *args):
+    if not args:
+        # If time is not specified, try to use $SOURCE_DATE_EPOCH variable
+        # See https://wiki.debian.org/ReproducibleBuilds/TimestampsProposal
+        source_date_epoch = os.getenv('SOURCE_DATE_EPOCH')
+        if source_date_epoch is not None:
+            time_struct = time.gmtime(float(source_date_epoch))
+            args = [time_struct]
+    if PY2:
         # if a locale is set, the time strings are encoded in the encoding
         # given by LC_TIME; if that is available, use it
         enc = locale.getlocale(locale.LC_TIME)[1] or 'utf-8'
         return time.strftime(text_type(format).encode(enc), *args).decode(enc)
-else:
-    ustrftime = time.strftime
+    else:
+        return time.strftime(format, *args)
 
 
 def safe_relpath(path, start=None):