Ten AWG steps is approximately ten times the cross sectional area. (e.g. from AWG 40 0.00501mm² to AWG 30 0.0509mm²).
When rewound on the same bobbin, there is room for 1/10 the number of turns, and requires 1/10 the wire length.
The resistance is 1/100 of the original (10x cross section, 1/10 length).
The inductance is also approximately 1/100 of the original, since on equal cores the inductance is as the square of the number of turns.
(I have used 10:1 because it yields 22-24V, but the ratios can be calculated for any gauge wire.)
When 1/10 the voltage is applied, the current is 10x, leaving the power dissipation approximately the same.
You should be able to estimate the gauge of the existing wire by carefully measuring the bobbin dimensions and the DC resistance of the coil. You could alternatively measure the length of the wire (after unwinding) and either the DC resistance or the bobbin size. As an example, I estimate that an 11,000 ohm coil on a 25mm long, 10 mm inside diameter, 20 mm outside diameter bobbin is approximately 51400 turns using 3.1 km of AWG 40. (If that coil was originally 240V, AWG 30 would be used to convert it to 24V. It would use about 300 m of enamel AWG 30 insulated wire.)
Sources of error:
--Insulation thickness is not accounted for. Enamel insulation is negligible with larger wire but becomes significant where the wire diameter is a few micrometers.
--DC resistance is a factor of temperature. Also assumes wire hasn't been stretched.
--Heavier gauge wire may not pack as efficiently.