diff --git a/pandas/tests/copy_view/test_methods.py b/pandas/tests/copy_view/test_methods.py
index 8fd9d5c5126c1..0cec5522e39cd 100644
--- a/pandas/tests/copy_view/test_methods.py
+++ b/pandas/tests/copy_view/test_methods.py
@@ -9,6 +9,7 @@
     Series,
     Timestamp,
     date_range,
+    period_range,
 )
 import pandas._testing as tm
 from pandas.tests.copy_view.util import get_array
@@ -53,6 +54,78 @@ def test_copy_shallow(using_copy_on_write):
         assert np.shares_memory(get_array(df_copy, "a"), get_array(df, "a"))
 
 
+@pytest.mark.parametrize("copy", [True, None, False])
+@pytest.mark.parametrize(
+    "method",
+    [
+        lambda df, copy: df.rename(columns=str.lower, copy=copy),
+        lambda df, copy: df.reindex(columns=["a", "c"], copy=copy),
+        lambda df, copy: df.reindex_like(df, copy=copy),
+        lambda df, copy: df.set_axis(["a", "b", "c"], axis="index", copy=copy),
+        lambda df, copy: df.rename_axis(index="test", copy=copy),
+        lambda df, copy: df.rename_axis(columns="test", copy=copy),
+        # lambda df, copy: df.astype({'b': 'int64'}, copy=copy),
+        # lambda df, copy: df.swaplevel(0, 0, copy=copy),
+        lambda df, copy: df.swapaxes(0, 0, copy=copy),
+        lambda df, copy: df.truncate(0, 5, copy=copy),
+        # lambda df, copy: df.infer_objects(copy=copy)
+        lambda df, copy: df.to_timestamp(copy=copy),
+        lambda df, copy: df.to_period(freq="D", copy=copy),
+        lambda df, copy: df.tz_localize("US/Central", copy=copy),
+        lambda df, copy: df.tz_convert("US/Central", copy=copy),
+        lambda df, copy: df.set_flags(allows_duplicate_labels=False, copy=copy),
+    ],
+    ids=[
+        "rename",
+        "reindex",
+        "reindex_like",
+        "set_axis",
+        "rename_axis0",
+        "rename_axis1",
+        # "astype",  # CoW not yet implemented
+        # "swaplevel",  # only series
+        "swapaxes",
+        "truncate",
+        # "infer_objects",  # CoW not yet implemented
+        "to_timestamp",
+        "to_period",
+        "tz_localize",
+        "tz_convert",
+        "set_flags",
+    ],
+)
+def test_methods_copy_keyword(
+    request, method, copy, using_copy_on_write, using_array_manager
+):
+    index = None
+    if "to_timestamp" in request.node.callspec.id:
+        index = period_range("2012-01-01", freq="D", periods=3)
+    elif "to_period" in request.node.callspec.id:
+        index = date_range("2012-01-01", freq="D", periods=3)
+    elif "tz_localize" in request.node.callspec.id:
+        index = date_range("2012-01-01", freq="D", periods=3)
+    elif "tz_convert" in request.node.callspec.id:
+        index = date_range("2012-01-01", freq="D", periods=3, tz="Europe/Brussels")
+
+    df = DataFrame({"a": [1, 2, 3], "b": [4, 5, 6], "c": [0.1, 0.2, 0.3]}, index=index)
+    df2 = method(df, copy=copy)
+
+    share_memory = (using_copy_on_write and copy is not True) or copy is False
+
+    if request.node.callspec.id.startswith("reindex-"):
+        # TODO copy=False without CoW still returns a copy in this case
+        if not using_copy_on_write and not using_array_manager and copy is False:
+            share_memory = False
+        # TODO copy=True with CoW still returns a view
+        if using_copy_on_write:
+            share_memory = True
+
+    if share_memory:
+        assert np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+    else:
+        assert not np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+
+
 # -----------------------------------------------------------------------------
 # DataFrame methods returning new DataFrame using shallow copy