I don’t think it necessary has to do with Cost of Capital increasing after a certain $ amount, but it has more to do with the leverage ratio and the balance sheet strength of the organization.
Let’s take a practical view here.
Let’s say you are a lender to a company that currently has no debt and Free Cash Flow of $10m for the year. Would you be comfortable lending that company $10m at LIBOR + 1%? You probably would because they will likely be able to pay you back quickly. However, let’s say that same company already had $100m in debt. Would you be comfortable lending that same company $10m in additional funds at LIBOR + 1%? Probably not, as you would demand a higher premium because the company has a higher risk of default and may not be able to repay the loan.
Likewise for shareholders, if a company is more leverage, the company has a higher risk and should be reflected in the volatility of the share price (ie. higher Beta) resulting in shareholders demanding a higher equity premium for investing, resulting in higher implied cost of equity.