> if we dole out UBI to everyone to cover basic living costs, will that simply result in the cost of rent going up to absorb the whole amount?

For that to be true, rent-seeking would have to literally capture all surplus. In which case UBI wouldn’t be an option in the first place.

The marginal increase in the purchasing power of someone who went from having $0 to $n would always be greater than the increase in purchasing power for someone who went from $1 billion to $1 billion + n - even with inflation.