If a district is worth one gold. Can the warlord destroy it without paying?
This is explicitly stated in the rules.
You receive one gold for each military (red) district in your city. At the end of your turn, you may destroy one district of your choice by paying a number of gold equal to one less than the cost of the district. Thus, you may destroy a cost one district for free, a cost two district for one gold, or a cost six district for five gold, etc. You may destroy one of your own districts. You may not, however, destroy a district in a city that is already completed by having eight districts (or seven districts when the Bell Tower is in play).