Before you do that, have you checked the battery? Cold weather will
sap a battery that's marginal and near the end of it's life (please
note it may have enough juice to power up the indicator
lights/buzzers/etc, but still may not turn the starter). Relays are
usually affected by heat, not a lack of it, so that wouldn't be my
first thing to look at (it's not impossible, but it's not my first
suspect).