[antognini]

No, the temperature differential is necessary. If life is consuming some food in a high energy state, converting it to a low energy state, and then using the difference for itself, it needs to put that waste energy somewhere once it's done with it. It can do this if the bath is at a lower temperature. But if the bath is at the higher temperature (i.e., if there's no differential) then the organism will be unable to convert the food into the lower energy state.

Strictly speaking, the important thing is to have a differential in entropy. The processes of life produce entropy, and that entropy has to go somewhere else if the organism is to live. But a differential in entropy implies a differential in temperature (though you may have to be careful about how you define temperature since there are different kinds of temperature).

[est31]

Consider everything in the universe being 40 deg C, including all matter, the CMB, etc. Consider now an organism that eats some food and releases the energy inside that food. It converts this energy into heat and i now 42 deg C. Suddenly there is a differential again. Sure, it can get rid of the heat easier if the differential were larger, but it can get rid of it at all, due to the existence of a differential.