Life tends to be wonderful and varying. Perhaps the example I cite below seems not realistic… but I assure you cases like this arise “all the time”:
There are situations where equity in some asset has essentially no carrying cost; in such situations, then the cost of equity will be less than the cost of debt, because debt *always* has some cost— even if you can secure an interest-free loan, that just means that your lender (usually a relative in such a case) is bearing the cost of the debt—by foregoing the return that he could have otherwise gotten on the money he lent you, at least.
What is an example of an asset with no carrying cost? Well suppose that you own free and clear, a share of some land, located in an area where there is no real estate tax. Then it costs
you nothing to carry the equity in that land.
If it strains credulity too much to imagine land not taxed, then simply substitute an asset such as mineral rights. Suppose that you have 50% equity in the mineral rights on some land that your brother inherited from your father. Further suppose that oil is discovered on closely adjoining land. Suddenly your equity in those mineral rights has real asset value, and the cost of equity for you has been zero, and will continue to be zero.