diff --git a/doc/source/whatsnew/v3.0.0.rst b/doc/source/whatsnew/v3.0.0.rst
index 1f42b75c5c5d2..88d36439af1d5 100644
--- a/doc/source/whatsnew/v3.0.0.rst
+++ b/doc/source/whatsnew/v3.0.0.rst
@@ -274,6 +274,7 @@ Performance improvements
 - Performance improvement in :meth:`MultiIndex.equals` for equal length indexes (:issue:`56990`)
 - Performance improvement in :meth:`RangeIndex.__getitem__` with a boolean mask or integers returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57588`)
 - Performance improvement in :meth:`RangeIndex.append` when appending the same index (:issue:`57252`)
+- Performance improvement in :meth:`RangeIndex.round` returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57824`)
 - Performance improvement in :meth:`RangeIndex.join` returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57651`, :issue:`57752`)
 - Performance improvement in :meth:`RangeIndex.reindex` returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57647`, :issue:`57752`)
 - Performance improvement in :meth:`RangeIndex.take` returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57445`, :issue:`57752`)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 0c955dc978cb8..c2df773326dc9 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -6737,7 +6737,6 @@ def diff(self, periods: int = 1) -> Index:
         """
         return Index(self.to_series().diff(periods))
 
-    @final
     def round(self, decimals: int = 0) -> Self:
         """
         Round each value in the Index to the given number of decimals.
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index 63fcddd961e04..8199ba8ed3a71 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -1165,6 +1165,42 @@ def any(self, *args, **kwargs) -> bool:
 
     # --------------------------------------------------------------------
 
+    # error: Return type "RangeIndex | Index" of "round" incompatible with
+    # return type "RangeIndex" in supertype "Index"
+    def round(self, decimals: int = 0) -> Self | Index:  # type: ignore[override]
+        """
+        Round each value in the Index to the given number of decimals.
+
+        Parameters
+        ----------
+        decimals : int, optional
+            Number of decimal places to round to. If decimals is negative,
+            it specifies the number of positions to the left of the decimal point
+            e.g. ``round(11.0, -1) == 10.0``.
+
+        Returns
+        -------
+        Index or RangeIndex
+            A new Index with the rounded values.
+
+        Examples
+        --------
+        >>> import pandas as pd
+        >>> idx = pd.RangeIndex(10, 30, 10)
+        >>> idx.round(decimals=-1)
+        RangeIndex(start=10, stop=30, step=10)
+        >>> idx = pd.RangeIndex(10, 15, 1)
+        >>> idx.round(decimals=-1)
+        Index([10, 10, 10, 10, 10], dtype='int64')
+        """
+        if decimals >= 0:
+            return self.copy()
+        elif self.start % 10**-decimals == 0 and self.step % 10**-decimals == 0:
+            # e.g. RangeIndex(10, 30, 10).round(-1) doesn't need rounding
+            return self.copy()
+        else:
+            return super().round(decimals=decimals)
+
     def _cmp_method(self, other, op):
         if isinstance(other, RangeIndex) and self._range == other._range:
             # Both are immutable so if ._range attr. are equal, shortcut is possible
diff --git a/pandas/tests/indexes/ranges/test_range.py b/pandas/tests/indexes/ranges/test_range.py
index 3040b4c13dc17..635812dcdd9fe 100644
--- a/pandas/tests/indexes/ranges/test_range.py
+++ b/pandas/tests/indexes/ranges/test_range.py
@@ -608,6 +608,37 @@ def test_range_index_rsub_by_const(self):
         tm.assert_index_equal(result, expected)
 
 
+@pytest.mark.parametrize(
+    "rng, decimals",
+    [
+        [range(5), 0],
+        [range(5), 2],
+        [range(10, 30, 10), -1],
+        [range(30, 10, -10), -1],
+    ],
+)
+def test_range_round_returns_rangeindex(rng, decimals):
+    ri = RangeIndex(rng)
+    expected = ri.copy()
+    result = ri.round(decimals=decimals)
+    tm.assert_index_equal(result, expected, exact=True)
+
+
+@pytest.mark.parametrize(
+    "rng, decimals",
+    [
+        [range(10, 30, 1), -1],
+        [range(30, 10, -1), -1],
+        [range(11, 14), -10],
+    ],
+)
+def test_range_round_returns_index(rng, decimals):
+    ri = RangeIndex(rng)
+    expected = Index(list(rng)).round(decimals=decimals)
+    result = ri.round(decimals=decimals)
+    tm.assert_index_equal(result, expected, exact=True)
+
+
 def test_reindex_1_value_returns_rangeindex():
     ri = RangeIndex(0, 10, 2, name="foo")
     result, result_indexer = ri.reindex([2])