Can I monitor the status of an appliance while it is on?

I have an electrical appliance which is switched on and off using a power-plug.
It is switched on and off on a schedule but can also be triggered manually.

The power plug supports power monitoring.
I know the appliance draws 600 watts.

I want to make an automation that monitors the appliance and alerts me when it is using an unexpected amount of power - only when it’s on.

So when it uses, say 0 or 300 when it’s on, something is probably wrong.
Same when it uses >1000.

But any automation will only be triggered ONCE when the device switches on or off.
How can I monitor it continuously?

But any automation will only be triggered ONCE when the device switches on or off.

Wrong. Above is true only if you select switch state as trigger. But, if you create an automation where trigger will be power consumption sensor it will trigger each time when power will be greater than, say 600W.

You can also create a template sensor like this:

- name: your sensor name
  state: >
    {%if states('sensor.yourdevice_power')|int >200%}
      STATE1
    {%elif states('sensor.yourdevice_power')|int >90%}
      STATE2 
    {%elif states('sensor.yourdevice_power')|int > 30%} 
      STATE3          
    {%else%}
      STATE4
    {% endif %}
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Thanks!
In the meantime, I found a helper ‘treshold’ that can also do this.
Same as in your suggestion, the key is to have a trigger when the power goes out of bounds and then check if the devices is switched on.

I use above sensor for my central heating stove (pellets) so i know when it’s starting to fire pellets, burning, cleaning or not working (error). Template sensors are extremely useful thing in HA.
Automation has trigger option “numeric state” and there you can define to trigger when power is above or below certain state. I think that you can even define multiple triggers and multiple actions depending on which trigger has been in action.