Fix mqtt subscribe debouncer initial delay too long when birth message is disabled (#92188)

Fix mqtt subscribe deboucer initial delay

homeassistant/components/mqtt/client.py

@@ -740,6 +740,9 @@ class MQTT:
             asyncio.run_coroutine_threadsafe(
                 publish_birth_message(birth_message), self.hass.loop
             )
+        else:
+            # Update subscribe cooldown period to a shorter time
+            self._subscribe_debouncer.set_timeout(SUBSCRIBE_COOLDOWN)
 
     async def _async_resubscribe(self) -> None:
         """Resubscribe on reconnect."""

tests/components/mqtt/test_init.py

@@ -2062,6 +2062,19 @@ async def test_no_birth_message(
         await asyncio.sleep(0.2)
         mqtt_client_mock.publish.assert_not_called()
 
+    async def callback(msg: ReceiveMessage) -> None:
+        """Handle birth message."""
+
+    # Assert the subscribe debouncer subscribes after
+    # about SUBSCRIBE_COOLDOWN (0.1) sec
+    # but sooner than INITIAL_SUBSCRIBE_COOLDOWN (1.0)
+
+    mqtt_client_mock.reset_mock()
+    await mqtt.async_subscribe(hass, "homeassistant/some-topic", callback)
+    await hass.async_block_till_done()
+    await asyncio.sleep(0.2)
+    mqtt_client_mock.subscribe.assert_called()
+
 
 @pytest.mark.parametrize(
     "mqtt_config_entry_data",