LG ThinQ Setup Failure

### The problem

Integration is failing to set up with error Failed setup, will retry: Failed to connect MQTT: ThinQAPIException: NOT_ALLOWED_API_AGAIN (1309) - Not allowed api call

### What version of Home Assistant Core has the issue?

2026.9.1

### What was the last working version of Home Assistant Core?

N?A

### What type of installation are you running?

Home Assistant OS

### Integration causing the issue

LG ThinQ

### Link to integration documentation on our website

https://www.home-assistant.io/integrations/lg_thinq

### Diagnostics information

ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.select has already been setup!
2026-09-07 12:01:13.553 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for sensor
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.sensor has already been setup!
2026-09-07 12:01:13.554 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for switch
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.switch has already been setup!
2026-09-07 12:01:13.555 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for vacuum
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.vacuum has already been setup!
2026-09-07 12:01:13.556 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for water_heater
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.water_heater has already been setup!
2026-09-07 12:11:14.446 WARNING (MainThread) [homeassistant.components.lg_thinq] No devices registered with the correct profile
2026-09-07 12:11:14.446 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for binary_sensor
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.binary_sensor has already been setup!
2026-09-07 12:11:14.448 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for climate
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.climate has already been setup!
2026-09-07 12:11:14.449 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for event
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.event has already been setup!
2026-09-07 12:11:14.450 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for fan
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.fan has already been setup!
2026-09-07 12:11:14.452 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for humidifier
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.humidifier has already been setup!
2026-09-07 12:11:14.453 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for number
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.number has already been setup!
2026-09-07 12:11:14.454 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for select
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.select has already been setup!
2026-09-07 12:11:14.455 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for sensor
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.sensor has already been setup!
2026-09-07 12:11:14.457 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for switch
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.switch has already been setup!
2026-09-07 12:11:14.458 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for vacuum
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.vacuum has already been setup!
2026-09-07 12:11:14.459 ERROR (MainThread) [homeassistant.components.lg_thinq] Error setting up entry LG ThinQ for water_heater

### Example YAML snippet

```yaml

```

### Anything in the logs that might be useful for us?

```txt
Logger: homeassistant.components.lg_thinq
Source: config_entries.py:803
Integration: LG ThinQ (documentation, issues)
First occurred: 6 September 2026 at 13:38:15 (1628 occurrences)
Last logged: 12:21:15

Error setting up entry LG ThinQ for select
Error setting up entry LG ThinQ for sensor
Error setting up entry LG ThinQ for switch
Error setting up entry LG ThinQ for vacuum
Error setting up entry LG ThinQ for water_heater
Traceback (most recent call last):
File "/usr/src/homeassistant/homeassistant/config_entries.py", line 803, in __async_setup_with_context
result = await component.async_setup_entry(hass, self)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/src/homeassistant/homeassistant/components/binary_sensor/__init__.py", line 137, in async_setup_entry
return await hass.data[DATA_COMPONENT].async_setup_entry(entry)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/src/homeassistant/homeassistant/helpers/entity_component.py", line 183, in async_setup_entry
raise ValueError(
...<2 lines>...
)
ValueError: Config entry LG ThinQ (01M1VBKS245SFG46BMBGC5KT4R) for lg_thinq.binary_sensor has already been setup!
```

### Additional information

Start with the LG ThinQ integration entry points under homeassistant/components/lg_thinq and the config_entries.py traceback showing repeated setup. Reproduce the setup failure and trace why the same config entry is initialized repeatedly across platforms. Done means the integration sets up once without the duplicate-setup errors and the reported MQTT failure is handled or resolved.

Written by the indexing model from the issue text.