diff --git a/dev/tasks/conda-recipes/arrow-cpp/meta.yaml b/dev/tasks/conda-recipes/arrow-cpp/meta.yaml
index e61034c3075..ac4b29eb5ee 100644
--- a/dev/tasks/conda-recipes/arrow-cpp/meta.yaml
+++ b/dev/tasks/conda-recipes/arrow-cpp/meta.yaml
@@ -403,8 +403,6 @@ outputs:
         {% set tests_to_skip = tests_to_skip + " or test_safe_cast_from_float_with_nans_to_int" %}  # [ppc64le]
         # gandiva tests are segfaulting on ppc
         {% set tests_to_skip = tests_to_skip + " or test_float_with_null_as_integer" %}   # [ppc64le]
-        # "Unsupported backend 'nonexistent' specified in ARROW_DEFAULT_MEMORY_POOL"
-        {% set tests_to_skip = tests_to_skip + " or (test_memory and test_env_var)" %}    # [unix]
         # test is broken; header is in $PREFIX, not $SP_DIR
         {% set tests_to_skip = tests_to_skip + " or (test_misc and test_get_include)" %}  # [unix]
         # flaky tests that fail occasionally
diff --git a/python/pyarrow/tests/test_memory.py b/python/pyarrow/tests/test_memory.py
index 092c50de33b..d9fdeb152c3 100644
--- a/python/pyarrow/tests/test_memory.py
+++ b/python/pyarrow/tests/test_memory.py
@@ -134,8 +134,14 @@ def check_env_var(name, expected, *, expect_warning=False):
         res.check_returncode()  # fail
     errlines = res.stderr.splitlines()
     if expect_warning:
-        assert len(errlines) == 1
-        assert f"Unsupported backend '{name}'" in errlines[0]
+        assert len(errlines) in (1, 2)
+        if len(errlines) == 1:
+            # ARROW_USE_GLOG=OFF
+            assert f"Unsupported backend '{name}'" in errlines[0]
+        else:
+            # ARROW_USE_GLOG=ON
+            assert "InitGoogleLogging()" in errlines[0]
+            assert f"Unsupported backend '{name}'" in errlines[1]
     else:
         assert len(errlines) == 0