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That is what happens with heaters. The resistance is much lower at room temperature than at (red-hot) operating temperature. The heater voltage is 12.6 V or 6.3 V, not 6.0 V.

The comments in the 12AU7heater model indicate that it works with both DC and AC so I gave that a whirl. 6VRMS (4.2V peak) actually gives time varying results. Current starts at a little over 500mA peak but settles down to around 200mA peak after a second or so.