More total costs divided by more total units is just the same cost per unit, if not lower because of economies of scale.
> Rents therefore increase, despite there being more supply.
If there is a monopoly landlord then rents will be at the monopoly rent whether you increase supply or not. Even then increasing supply could lower rents, because the monopoly landlord could capture more rents by charging $9000/month on twice as many units than $10,000/month on half as many units, and can't charge $10,000/month on twice as many units because there aren't enough tenants who can afford that.
Also, the premise here is that you're increasing supply. The monopoly landlord would have to outbid everybody else for the new supply or they'd lose their monopoly, and have to pay the monopoly price or else the new units would be cheaper than their existing ones. But then the construction companies would be receiving the monopoly price and become flush with cash to build even more housing until the monopoly landlord ran out of money.
There is a reason landlords collude through zoning boards: It's otherwise quite easy for someone new to enter the market.