Thank you for your comments BachelorDays and Fat Boy.
When I switch off the car and then restart it again when it is in the shaking mode it does not help. It keeps shaking until it has completely cooled down and only then restarted.
When it is cold or in normal operating temperature I can switch off and on and runs well.
I tried to google the Temp sensor but am not sure what part is that. Interesting about the throttle body and maybe will try to change this and see.
I don't have have belt but a chain - is it possible that the chain needs changing?
Many thanks again.
I ordered Portable Mini VCI Diagnostic Tool Cable Scanner for Toyota Series Auto Cars and will see if it gives me more details when connect to the computer for analysis...