**Build scan:**
https://gradle-enterprise.elastic.co/s/vsdqw2nozrf56/tests/:x-pack:plugin:frozen-indices:internalClusterTest/org.elasticsearch.index.engine.frozen.FrozenIndexTests/testSearchAndGetAPIsAreThrottled

**Reproduction line:**
```
./gradlew ':x-pack:plugin:frozen-indices:internalClusterTest' --tests "org.elasticsearch.index.engine.frozen.FrozenIndexTests.testSearchAndGetAPIsAreThrottled" -Dtests.seed=E85751702E4664BA -Dtests.locale=sr-Latn-RS -Dtests.timezone=America/Marigot -Druntime.java=20
```

**Applicable branches:**
main

**Reproduces locally?:**
No

**Failure history:**
https://gradle-enterprise.elastic.co/scans/tests?tests.container=org.elasticsearch.index.engine.frozen.FrozenIndexTests&tests.test=testSearchAndGetAPIsAreThrottled

**Failure excerpt:**
```
java.lang.AssertionError: no warnings, expected: [WARN: Searching frozen indices [index] is deprecated. Consider cold or frozen tiers in place of frozen indices. The frozen feature will be removed in a feature release.]

  at __randomizedtesting.SeedInfo.seed([E85751702E4664BA:37B829D30841FBC6]:0)
  at org.junit.Assert.fail(Assert.java:88)
  at org.junit.Assert.assertTrue(Assert.java:41)
  at org.junit.Assert.assertNotNull(Assert.java:712)
  at org.elasticsearch.test.ESTestCase.assertWarnings(ESTestCase.java:586)
  at org.elasticsearch.test.ESTestCase.assertWarnings(ESTestCase.java:555)
  at org.elasticsearch.index.engine.frozen.FrozenIndexTests.testSearchAndGetAPIsAreThrottled(FrozenIndexTests.java:219)
  at jdk.internal.reflect.DirectMethodHandleAccessor.invoke(DirectMethodHandleAccessor.java:104)
  at java.lang.reflect.Method.invoke(Method.java:578)
  at com.carrotsearch.randomizedtesting.RandomizedRunner.invoke(RandomizedRunner.java:1758)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$8.evaluate(RandomizedRunner.java:946)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$9.evaluate(RandomizedRunner.java:982)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$10.evaluate(RandomizedRunner.java:996)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at org.apache.lucene.tests.util.TestRuleSetupTeardownChained$1.evaluate(TestRuleSetupTeardownChained.java:48)
  at org.apache.lucene.tests.util.AbstractBeforeAfterRule$1.evaluate(AbstractBeforeAfterRule.java:43)
  at org.apache.lucene.tests.util.TestRuleThreadAndTestName$1.evaluate(TestRuleThreadAndTestName.java:45)
  at org.apache.lucene.tests.util.TestRuleIgnoreAfterMaxFailures$1.evaluate(TestRuleIgnoreAfterMaxFailures.java:60)
  at org.apache.lucene.tests.util.TestRuleMarkFailure$1.evaluate(TestRuleMarkFailure.java:44)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl$StatementRunner.run(ThreadLeakControl.java:390)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl.forkTimeoutingTask(ThreadLeakControl.java:843)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl$3.evaluate(ThreadLeakControl.java:490)
  at com.carrotsearch.randomizedtesting.RandomizedRunner.runSingleTest(RandomizedRunner.java:955)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$5.evaluate(RandomizedRunner.java:840)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$6.evaluate(RandomizedRunner.java:891)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$7.evaluate(RandomizedRunner.java:902)
  at org.apache.lucene.tests.util.AbstractBeforeAfterRule$1.evaluate(AbstractBeforeAfterRule.java:43)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at org.apache.lucene.tests.util.TestRuleStoreClassName$1.evaluate(TestRuleStoreClassName.java:38)
  at com.carrotsearch.randomizedtesting.rules.NoShadowingOrOverridesOnMethodsRule$1.evaluate(NoShadowingOrOverridesOnMethodsRule.java:40)
  at com.carrotsearch.randomizedtesting.rules.NoShadowingOrOverridesOnMethodsRule$1.evaluate(NoShadowingOrOverridesOnMethodsRule.java:40)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at org.apache.lucene.tests.util.TestRuleAssertionsRequired$1.evaluate(TestRuleAssertionsRequired.java:53)
  at org.apache.lucene.tests.util.AbstractBeforeAfterRule$1.evaluate(AbstractBeforeAfterRule.java:43)
  at org.apache.lucene.tests.util.TestRuleMarkFailure$1.evaluate(TestRuleMarkFailure.java:44)
  at org.apache.lucene.tests.util.TestRuleIgnoreAfterMaxFailures$1.evaluate(TestRuleIgnoreAfterMaxFailures.java:60)
  at org.apache.lucene.tests.util.TestRuleIgnoreTestSuites$1.evaluate(TestRuleIgnoreTestSuites.java:47)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl$StatementRunner.run(ThreadLeakControl.java:390)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl.lambda$forkTimeoutingTask$0(ThreadLeakControl.java:850)
  at java.lang.Thread.run(Thread.java:1623)

```
x-pack/plugin/frozen-indices/src/internalClusterTest/java/org/elasticsearch/index/engine/frozen/FrozenIndexTests.java
() -> client().prepareIndex(indexName).setId("4").setSource("field", "value").setRefreshPolicy(IMMEDIATE).get()
        );
        IndicesService indexServices = getInstanceFromNode(IndicesService.class);
        Index index = resolveIndex(indexName);
        IndexService indexService = indexServices.indexServiceSafe(index);
        IndexShard shard = indexService.getShard(0);
        Engine engine = IndexShardTestCase.getEngine(shard);
        assertEquals(0, shard.refreshStats().getTotal());
        boolean useDFS = randomBoolean();
        assertHitCount(
            client().prepareSearch()
                .setIndicesOptions(IndicesOptions.STRICT_EXPAND_OPEN_FORBID_CLOSED)
                .setSearchType(useDFS ? SearchType.DFS_QUERY_THEN_FETCH : SearchType.QUERY_THEN_FETCH)
                .get(),
            3
        );
        assertThat(engine, Matchers.instanceOf(FrozenEngine.class));
        assertEquals(useDFS ? 3 : 2, shard.refreshStats().getTotal());
        assertFalse(((FrozenEngine) engine).isReaderOpen());
        assertTrue(indexService.getIndexSettings().isSearchThrottled());
        
        SearchResponse searchResponse = client().prepareSearch()
            .setIndicesOptions(IndicesOptions.STRICT_EXPAND_OPEN_FORBID_CLOSED)
            .setScroll(TimeValue.timeValueMinutes(1))
            .setSize(1)
            .get();
        do {
            assertHitCount(searchResponse, 3);
            assertEquals(1, searchResponse.getHits().getHits().length);
            SearchService searchService = getInstanceFromNode(SearchService.class);