Tungsten has a positive temperature coefficient of resistance, so a cold
lamp filament has a much lower resistance than it does at operating
temperature. The reason one sometimes sees a lower "Tungsten" rating
specified for switches and relays is because when an incandescent lamp
is turned on it results in a high initial surge current until the
filament heats up.

John

Actually that is bass ackwards
Tungsten filaments have a negative temperature coefficient.
Temperature rises conductivity decreases.
Back when semiconductors were young a lamp was often used to help
prevent thermal runaway in amplifier circuits by placing them in
series with the transistor.

In any case most lamp timers should be able to handle a 400w metal
halide set.
Just make sure it is a grounded timer ( 3 prong)
Better still would be a grounded timer on a GFCI protected circuit.