^^^ Yep, temp sensor almost certainly.
Mine suddenly failed recently too in the same way. Engine was getting to temperature but the gauge wasn't moving.
Just make sure when you change it that the crappy plastic retaining clip is properly secured, if you don't want to spend a few hours on the hard shoulder like I did:
http://forum.a8parts.co.uk/showthread.php?t=9845
Even if the clip seems to be properly seated, I'd recommend dropping some heat-shrink tubing over it to prevent it sliding out.
PS: It's at the back of the engine. Might take a lot of wiggling and swearing to get the old one out.
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Mark
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2002 FE S8 Ebony Black Pearl
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Cars Owned:
The Tesla Era: 2020 Model S Performance Ludicrous+ (present)
(Black, with all black premium interior and carbon fibre décor, 21" sonic carbon twin turbine wheels and FSD capability)
The Audi Era: '97 A8 4.2 (Ming Blue) --> '96 A8 4.2 QS (Dark Green) --> '02 FE S8 (present)
The Citroen Era: '84 BX 1.6 RS --> '89 BX 1.9 DTR Turbo --> '94 XM 2.0L Turbo --> '96 XM 2.0L Turbo Exclusive --> '00 Xantia Activa 2.0L Turbo
The Banger Era: '76 1.2L Lada VAZ-2101 (Ruski Fiat 124) --> '80 1.7L Morris Ital HL, finished in Ermine White and Rust