 Whether or not a tide-locked planet with the orbital characteristics of Gliese 581g is actually habitable depends on the composition of the atmosphere and the nature of the planetary surface. A comprehensive modeling study including atmospheric dynamics, realistic radiative transfer and the physics of formation of sea ice (if the planet has an ocean) indicates that the planet can become as hot as Venus if it is dry and allows carbon dioxide to accumulate in its atmosphere. The same study identified two habitable states for a water-rich planet. If the planet has a very thin atmosphere, a thick ice crust forms over most of the surface, but the substellar point remains hot enough to yield a region of thin ice or even episodically open water. If the planet has an atmosphere with Earthlike pressures, containing approximately 20 % (molar) carbon dioxide, then the greenhouse effect is sufficiently strong to maintain a pool of open water under the substellar point with temperatures comparable to the Earth's tropics. This state has been dubbed "Eyeball Earth" by the author.
