diff --git a/asv_bench/benchmarks/index_object.py b/asv_bench/benchmarks/index_object.py
index 9c05019c70396..2b2302a796730 100644
--- a/asv_bench/benchmarks/index_object.py
+++ b/asv_bench/benchmarks/index_object.py
@@ -86,6 +86,12 @@ def time_iter_dec(self):
         for _ in self.idx_dec:
             pass
 
+    def time_sort_values_asc(self):
+        self.idx_inc.sort_values()
+
+    def time_sort_values_des(self):
+        self.idx_inc.sort_values(ascending=False)
+
 
 class IndexEquals:
     def setup(self):
diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index 17b9e49bcad6a..07d1d39ec4221 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -427,6 +427,7 @@ Performance improvements
 - Performance improvement in :meth:`SparseArray.take` with ``allow_fill=False`` (:issue:`43654`)
 - Performance improvement in :meth:`.Rolling.mean` and :meth:`.Expanding.mean` with ``engine="numba"`` (:issue:`43612`)
 - Improved performance of :meth:`pandas.read_csv` with ``memory_map=True`` when file encoding is UTF-8 (:issue:`43787`)
+- Performance improvement in :meth:`RangeIndex.sort_values` overriding :meth:`Index.sort_values` (:issue:`43666`)
 - Performance improvement in :meth:`RangeIndex.insert` (:issue:`43988`)
 - Performance improvement in :meth:`Index.insert` (:issue:`43953`)
 - Performance improvement in :meth:`DatetimeIndex.tolist` (:issue:`43823`)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index c5cfa4c2d1fcd..05047540c6ccd 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -5266,7 +5266,6 @@ def asof_locs(self, where: Index, mask: np.ndarray) -> npt.NDArray[np.intp]:
 
         return result
 
-    @final
     def sort_values(
         self,
         return_indexer: bool = False,
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index 4def2e4b93553..c5ba928a09cfa 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -522,6 +522,38 @@ def equals(self, other: object) -> bool:
             return self._range == other._range
         return super().equals(other)
 
+    def sort_values(
+        self,
+        return_indexer: bool = False,
+        ascending: bool = True,
+        na_position: str = "last",
+        key: Callable | None = None,
+    ):
+        sorted_index = self
+        indexer = RangeIndex(range(len(self)))
+        if key is not None:
+            return super().sort_values(
+                return_indexer=return_indexer,
+                ascending=ascending,
+                na_position=na_position,
+                key=key,
+            )
+        else:
+            sorted_index = self
+            if ascending:
+                if self.step < 0:
+                    sorted_index = self[::-1]
+                    indexer = indexer[::-1]
+            else:
+                if self.step > 0:
+                    sorted_index = self[::-1]
+                    indexer = indexer = indexer[::-1]
+
+        if return_indexer:
+            return sorted_index, indexer
+        else:
+            return sorted_index
+
     # --------------------------------------------------------------------
     # Set Operations
 
diff --git a/pandas/tests/indexes/ranges/test_range.py b/pandas/tests/indexes/ranges/test_range.py
index 7591620de168a..d58dff191cc73 100644
--- a/pandas/tests/indexes/ranges/test_range.py
+++ b/pandas/tests/indexes/ranges/test_range.py
@@ -607,3 +607,11 @@ def test_isin_range(self, base):
         result = base.isin(values)
         expected = np.array([True, False])
         tm.assert_numpy_array_equal(result, expected)
+
+    def test_sort_values_key(self):
+        # GH#43666
+        sort_order = {8: 2, 6: 0, 4: 8, 2: 10, 0: 12}
+        values = RangeIndex(0, 10, 2)
+        result = values.sort_values(key=lambda x: x.map(sort_order))
+        expected = Index([4, 8, 6, 0, 2], dtype="int64")
+        tm.assert_index_equal(result, expected, check_exact=True)
diff --git a/pandas/tests/test_algos.py b/pandas/tests/test_algos.py
index 74bcb589b008a..8f0991eb98bb5 100644
--- a/pandas/tests/test_algos.py
+++ b/pandas/tests/test_algos.py
@@ -68,7 +68,7 @@ def test_factorize(self, index_or_series_obj, sort):
         expected_codes = np.asarray(expected_codes, dtype=np.intp)
 
         tm.assert_numpy_array_equal(result_codes, expected_codes)
-        tm.assert_index_equal(result_uniques, expected_uniques)
+        tm.assert_index_equal(result_uniques, expected_uniques, exact=True)
 
     def test_series_factorize_na_sentinel_none(self):
         # GH#35667