Sorry I meant to say that if all houses double in value, and municipal costs don't change, then your taxes do not change. Just because the value goes up does not imply that taxes go up.
FWIW, I live in Texas so this place gets a good chunk of its tax revenue from property taxes. The odds of them cutting the tax rate just because your house shot up in cost at least to me seems pretty low. This is one of the problems of gentrification, where long time owners find they are unable to continue owning in the $90k house they bought 20 years ago because now they pay property taxes on a $750k property.