I’m actually a bit surprised not to have found this little gem on E2 before. I know of several versions of the story floating around on the internet, but this one is adapted by me from the original – as far as I know – source: Applied Optics, II, A14 (1972), written by anon.
Here we go:
Proof that Heaven is Hotter than Hell
The temperature of
heaven can be rather accurately computed. Our authority is the
Bible, more exactly
Isaiah 30:26, which states:
Moreover, the light of the moon shall be as the light of the sun and the light of the sun shall be sevenfold as the light of seven days.
Therefore, heaven receives from the moon as much radiation as the earth does from the sun, and in addition seven times seven (forty nine) times as much as the earth does from the sun, or fifty times in all. The light we receive from the moon is one tenthousandth of the light we receive from the sun, so we can ignore that. With these data we can compute the temperature of heaven: The radiation falling on heaven will heat it to the point where the heat lost by radiation is just equal to the heat received by radiation. In other words, heaven loses fifty times as much heat as the earth by radiation. Using the StefanBoltzmann fourth power law for radiation
(H/E)^{4} = 50
where E is the absolute temperature of the earth, 300K . This gives H the absolute temperature of heaven, as 798K absolute (525°C).
Now, we know from Revelations 21:8:
But the fearful and unbelieving... shall have their part in the lake which burneth with fire and brimstone.
A lake of molten brimstone – that’s sulphur, to you and me – means that its temperature must be at or below the boiling point, which is 444.6°C. (Above that point, it would be a vapor, not a lake.) The exact temperature of hell cannot be computed but it must be less than 444.6°C, the temperature at which brimstone or sulfur changes from a liquid to a gas.
So, we have:
Temperature of Heaven = 525°C
Temperature of Hell < 444.6°C.
Therefore Heaven is hotter than Hell...
... or is it?
Science hasn't got where it's got by sitting around all day. Oh no siree! Here is further thought, propagated by Dr. Tim Healey, F.R.C.R.; M.I.Nuc.E.
A refutation of the proof that Heaven is hotter than Hell
In Applied Optics (1972, 11 A14) there appeared a calculation of the respective temperatures of Heaven and Hell. That of Heaven was computed by substituting the values given in Isaiah 30:26 ^{1} in the StefanBoltzmann radiation law, so that (H/E)^{4} = 50, where E, the absolute temperature of the Earth, is 300K, whence the temperature of Heaven, H, is 798K or 525ºC. This is hard to find fault with.
The assessment of the temperature of Hell stands, I suggest, on less firm ground. As authority we use the data provided in Revelations 21:8 ^{2}, so that the temperature of Hell seems to be 444.6ºC – the temperature at which liquid sulfur is in equilibrium with its vapour – a temperature which is indeed sufficiently reliable to be used in the secondary calibration of pyrometers.
Now this last reckoning fails to follow the argument through. 444.6ºC is the temperature at which liquid sulphur is in equilibrium with its vapour at normal atmospheric pressure. Have we any data as to the pressure likely to be found in Hell?
The answer is yes.
A nineteenth century mathematician has already provided the groundwork for us ^{3} and we may feel confident that by the year 2000 the total number of the damned will be at least 29,422,641,251,519,917,000 souls. Yet the area of the valley of Gehinnom ^{4} is only 7,000,000 square meters.
We can now apply these figures in the Ideal Gas Equation to calculate what the pressure will be in the valley of Gehinnom. Since surely some souls must have been damned since 1877 ^{5}, the pressure can only have increased since these calculations were made and the equilibrium point on the phase diagram of sulfur must have shifted still further, so that if we can show that at a temperature of 525ºC sulfur would still be liquid at the pressure calculated (which is a minimum value, remember).
Hell is now cooler than Heaven.
Certain corrections must be applied first, however:

Neiht based his calculation on a date of creation of 1658 + 2326  1877 = 2107 BC (minimum). Counting generations in the Bible gives a date for the Creation of 4004 BC. However, atomic dating has shown that Olduvai man is at least 2 × 10^{6} years old ^{6}.

We should use a Fibonacci series for the expansion, not a simple doubling series. ^{7} The ancient Jewish laws against inbreeding also act in the same direction. ^{8}

By a fortunate coincidence, the effects of 1 and 2 cancel each other exactly. ^{9}

The human body is not an ideal gas, but...

A good deal of it is gaseous at 525ºC, and in any case;

It could well be that at very great pressures the external pressure may well exceed the pressure of electromagnetic repulsion, when different "gas" laws would apply. This merely explains how the Lord works in fitting so large a number of damned souls into so small a space ^{10} and it need not be quantitative.
In the calculation the following assumptions are made:

The average height can be taken as one meter. This seems a fair figure between the newborn babe and the fullgrown man.

The average space needed is about 30 cm × 20 cm. It seems unlikely that any closer packing could be achieved. Neiht uses a figure of 1/20 cubic meters per person, which is nearly identical with my independent assessment. Mine allows a neat cancellation, later.

I have assumed that not more than two layers of damned persons can be accommodated, since otherwise those in the middle layers would escape the full rigours of Hell.
So that
The volume available in Gehenna is 60 × 106 × 2 m^{3} and
The original volume of the damned is 0.06 × 29.422641 × 10^{18}m^{3}
Then, at constant temperature (which we assume, taking equilibrium)
P_{1}V_{1} = P_{2}V_{2} or P_{2} = P_{1}V_{1}/V_{2}
Substituting,
P_{2} = (29 × 6 × 1016) / (2 × 6 × 107) = 14.5 × 10^{9} atmospheres (1)
Now let us see what pressure is needed to liquefy sulphur vapour at 525ºC. We have, using the ClausiusCleypeyron equation in its integrated form
Log P = (7.43287  3268.2) / T
where P = pressure in mm Hg and T = the elevated boiling point in Kelvin,
so that
Log P = 7.43287  (3268.2/798) = 3.3373813
whence,
P = 2174.607 mm Hg = 2.86 atmospheres (2)
(1) is so much greater than (2) that Revelations 21:8 indicates a temperature very considerably higher than 525ºC.
Thus, Hell is hotter than Heaven (which remains deucedly hot).
Notes
(1) "The light of the moon shall be as the light of the sun and the light of the sun shall be sevenfold as the light of seven days..." The light of the moon is negligible in comparison with that of the sun.
(2) "...the lake which burneth with fire and brimstone: which is the second death".
(3) A Mathematical Proof of the NonExistence of Hell from the writings of the freethinker Neiht, born in Brussels, 1877. "The area of the valley of Jehoshaphat is 60,000,000 sq. ms. ... Supposing that each race originated with one couple only, one has five couples or ten people, and applying to them the principle of compound interest, up to the Flood there were 9,289,000 births in 1,658 years. Since the Flood up to our epoch 2,326 years have passed, during which, if only five couples survived, they would have produced 2,213,867,610,000 children. If these calculations are extended up to the year 2000, the resulting number is 34,326,414,259,675,172,000 which, together with the 9,289,000, makes 34,326,414,259,684,461,000 offspring. If one concedes, charitably, that all papists are saved, their number today being 1/7th of the population of the earth, that of the damned would be made up of those born before the Flood plus those born since the Flood up to the year 2000 minus the 1/7th of those born since the year 44, that of the birth of Christ: this number is 4,903,773,008,164,544,000, and the total of damned would be 29,422,641,251,519,917,000.
"The mean cubic area between a newborn infant and an adult is about 1/20th meter; the bulk of the damned above is equal to the mass of a sphere of radius 705,504 meters; that of the earth is 6,366,200 meters.
"If one puts back the origin of man, following certain German naturalists, to 80,000 years, the number of damned would form a cube three times the size of the earth.
"Now, how does one assemble the 34,326,414,260 millions risen on a surface of 60,000,000 sq. metres to judge them and how does one sink this mass of damned, through all manner of rock, to a depth of 5,660,660 metres?"
(4) The valley of Jehoshaphat is the Gehenna of the Jews (Jehennam in the Koran) – the place of eternal torment. The word is derived from Gehinnom = the valley of Hinnom where sacrifices were offered to Moloch (2 Corinthians 33:6) (= Adremmelech – the God of Sepharvaim). In later times, all manner of refuse was dumped there and fires were constantly maintained to consume it. The sulfurous stench and the fire was the original of the Christian concept of Hell. (The estimate of Nieht of Gehenna's area 60 × 106 sq. m. is wrong, according to measurements taken on the spot by the Editors).
(5) e.g. Dzhugashvili, Losif and DeSalvo, Albert H.
(6) Curtis G. H. and Evemden, J. F. in Nuclear Clocks USAEC Pp 4041.
(7) Leonardo da Pisa, Liber Abaci 1202 (Out of print).
(8) Leviticus 18:6 et seq.
(9) Westfall, R. S. Newton and the Fudge Factor. Science 1973, 751 758.
(10) "In my father's house are many mansions: If it were so I would have told you. I go to prepare a place for you." John 14:2. Also, "Those things which are impossible with men, are possible with God"  Luke 18:27, and "...with God all things are possible." Matthew 19:26.
Disclaimer: This paper appeared in The Journal of Irreproducible Results, Vol 25, No.4:1718
Copyright © 1979 by The Journal of Irreproducible Results, Inc.
I have it from: http://www.lhup.edu/~dsimanek/hell.htm