Earth, of course, has two (semi-permanently) frozen regions around the poles, with a very large habitable zone in between. Could a planet exist – habitable for humans – that had two livable zones around the poles, but a “boiling zone” at the equator, at which water reached 212[sup]o[/sup] Fahrenheit? Or would so hot a planet manifest a “Greenhouse Effect” and wind up like Venus?
Hmmm… Does the equatorial region on your planet have to be at the boiling point, or do you just want it hot enough for humans to want to stay out?
Also, any greenhouse effect (warming of a planet beyond that expected for a given level of incoming solar radiation) that comes into play is a function of the planet’s atmospheric composition. Can we assume a modern Earth atmosphere, or were you thinking of something a bit different?
The answer of course is “it depends”, on a lot of different factors. Even here on Earth, the average global temperature and climate has varied enormously over geologic time, due to the composition of the atmosphere and the distribution of the continents.
As a guess, I’d say what you’d want is a planet with a continuous equatoral landmass, with two circumpolar oceans. Islands in the oceans and two strips of land at high latitudes would be habitable, with soaring temperatures at the equator.
Well, I was thinking boiling, but I’d be interested in just extremely hot (say, 150[sup]o[/sup] F and up).
Yes, a modern Earth-like atmosphere only. Something humans could breathe.
I would think (this is a WAG) that if it’s actually hot enough to BOIL water at the equator, you’d definitely end up with so much water in the atmosphere that you’d get a runaway greenhouse effect unless you could think of some way to keep all the water away from the equator.
If you look at Venus, the temperature at the poles is just the same as the equator. Hardly any light manages to penetrate the thick clouds to reach the surface, so angle of incidence makes no difference. It’s all wrapped up in a nice, uniformly hot atmosphere. No wind near the surface, 'cause it’s the same all over.
I’d be hard pressed to determine exactly how warm is too warm. Maybe a really high axial tilt would help by exaggerating the difference between the pole and the equator, but go to far, and seasons get too extreme–you bake one pole and freeze the other.
Well, for starters, you’d need a planet with a rotational axis inclined not much more than Earth’s (about 23-26 deg. from the vertical), because you’d need most of the incoming solar radiation to strike at low latitudes.
Beyond that… as Lumpy noted, a whole bunch of factors come into play. With an atmospheric composition like modern Earth, you won’t reach average annual air temperatures near the boiling point, or probably even 150 deg. F, without bumping up the incoming solar radiation (or moving the planet into a closer orbit around a Sun-like star).
Without resorting to a change in orbit or solar output, it shouldn’t be too difficult to obtain daytime high temperatures of 150 deg. F or so, provided you had a sufficiently large land mass in the tropics; after all, a high temp. of 135 deg. F or so has already been recorded in the Sahara. However, night temps under desert conditions can be pretty cold; low humidity/lack of ground moisture means you’re not forming clouds to trap heat at night, so much of the heat radiates away.
It is possible to have a comfortable climate in the polar regions without the equatorial regions being outrageously (if uncomfortably) hot, especially if the tropics were largely land-free. In this case, the ocean would absorb much of the incoming heat and transport it poleward (think major Gulf Stream effect). The Cretaceous period is a good example of that type of scenario.
seeing Podkayne’s post on preview
In a scenario such as Lumpy described, with landmass concentrated at the equator (a ringworld configuration), you could possibly avoid injecting too much water vapor into the atmosphere (and accompanying greenhouse effects) over the long haul. An interesting side effect of having (relatively) high-albedo land in equatorial regions is that less heat enters the global climate system. The air temps might be ungodly hot for humans, but that heat would be largely re-radiated back into space. A concentration of landmass near the equator is currently thought to be a driving force behind the initiation of one of the worst glacial episodes in Earth’s history!
A high-obliquity world (more than 54 deg.) would be more likely to give the opposite result of what scratch intended - the poles would indeed alternately freeze and fry, and the equatorial regions would be the only “hospitable” areas.
I’m just not certain that you’d be able to achieve the scenario you describe, scratch, for one other reason: heat transport. Heat transport from the tropics to the poles is what drives our weather, and contributes to the energy balance of the climate system. In order to maintain boiling temps (on land) at the equator, and have comfortable (75 deg. F?) temps at the poles, I figure you’d need to find a way to set up and maintain a pretty steep temperature gradient somewhere in the mid-latitudes that winds and storms (and you’d have some hellacious ones) could not dispel. Any atmospheric physicist wants to come in here and explain to me how it could be done, I’d be happy to know. 
But my guess is it’s not possible without some fundamental changes to your model.
Harry Harrison describes a high-obliquity planet such as that which Fillet described in his book Wheelworld. I cannot recall the details exactly, but the upshot was that one hemisphere facing the sun was near boiling, while the twilight zone was habitable. The planet had a wobble, so every 20 years or so they would have to pack up and drive down a single highway to the other hemisphere in protecte vehicles, hence the “wheels.”
Well, let’s look at it in terms of present conditions. I haven’t tried to look up actual figures for average annual temperatures for various localities on Earth, but most for most places in the humid tropics they’re going to be around 80[sup]o[/sup] F. (Mid-latitude deserts get a lot hotter at times, but since they get colder at night the average temperature may not be that different.) In Antarctica summertime temperatures aren’t much above freezing even in coastal areas, so let’s say average annual temperature is 0[sup]o[/sup], but it could be as low as minus 30[sup]o[/sup].
So if we imagine raising average global temperatures uniformly by 80[sup]o[/sup], the poles would be around 80[sup]o[/sup], while the tropics would be at 160[sup]o[/sup]. Of course, there would be such massive changes in world meteorology and air circulation it would be hard to predict exactly how climate would change.
According to some paleoclimatologists, the Late Cretaceous was a “super-greenhouse” period in which the poles had sub-tropical conditions while the tropics may have reached an average temperature of 110[sup]o[/sup], IIRC.