diff --git a/src/pip/_internal/locations/_sysconfig.py b/src/pip/_internal/locations/_sysconfig.py
index 97aef1f1ac2..9dd802a3a22 100644
--- a/src/pip/_internal/locations/_sysconfig.py
+++ b/src/pip/_internal/locations/_sysconfig.py
@@ -1,3 +1,4 @@
+import dataclasses
 import logging
 import os
 import sys
@@ -192,9 +193,10 @@ def get_scheme(
         data=paths["data"],
     )
     if root is not None:
+        converted_keys = {}
         for key in SCHEME_KEYS:
-            value = change_root(root, getattr(scheme, key))
-            setattr(scheme, key, value)
+            converted_keys[key] = change_root(root, getattr(scheme, key))
+        scheme = dataclasses.replace(scheme, **converted_keys)
     return scheme
 
 
diff --git a/src/pip/_internal/models/scheme.py b/src/pip/_internal/models/scheme.py
index 4ca61fe0cce..06a9a550e34 100644
--- a/src/pip/_internal/models/scheme.py
+++ b/src/pip/_internal/models/scheme.py
@@ -10,7 +10,7 @@
 SCHEME_KEYS = ["platlib", "purelib", "headers", "scripts", "data"]
 
 
-@dataclass
+@dataclass(frozen=True)
 class Scheme:
     """A Scheme holds paths which are used as the base directories for
     artifacts associated with a Python package.