Why is radiation pressure dominant at high temperatures? In the ($\log\rho, \log T$) plane for stars, the lower right corner corresponds to the equation of state of radiation pressure. This means that as a star gets hotter and less dense, radiation pressure is dominant.
However, high temperatures lead to more particles and less photons by processes like photodisintegration and pair-production, lowering the radiation pressure.
How do these two facts fit?
I am guessing the answer is that more exothermic reactions happen at high temperatures than photodisintegration and pair-production reactions. Is this always true?
Also, why is low density needed? Why is there a maximum density for a fixed  temperature for which radiation pressure is dominant?
 A: See the equation and plot in pg. 17 of the attached. It's a simple derivation. http://www.astro.caltech.edu/~george/ay20/Ay20-Lec7x.pdf
The pressure of the matter is taken to be that of an ideal gas as function of density and temperature, linear with each $p = R/u * \rho * T$, where the radiation pressure comess from the blackbody isotropic radiation equation $p = a * T^4$, where p is pressure, a, R and u constants, T temperature and $\rho$ is density of the matter. 
You set the two pressures equal and see where is the dividing line. At higher temperature (more at higher left, with log T vertical), it's higher temperature and the radiation pressure dominates. Lower right is the opposite. 
If less matter is there and more photons the temperature will be higher, and vice versa down, so the equation gives you where each dominates, and is purely dependent on the density of matter and the equilibrium temperature. Most stars are in thermal equilibrium. Most of the time
A: It's true that photons can create electron-positron pairs. However, these photons must have quite a lot of energy - at least 1.022 MeV. This falls solidly into the energy range of gamma rays. Now, gamma rays may be produced in varying amounts in stars of all different masses, but they don't become significant sources of pair production until temperatures of $\sim10^9$ or $\sim10^{10}$ Kelvin, about 100 times as hot as the center of the Sun. Photodisintegration becomes a possibility at $\sim10^{10}$ Kelvin. Neither of these processes become quite as important in stars under about 100 solar masses.
Now, radiation pressure can become very important or even dominant over gas pressure at much lower temperatures and masses, often at $\sim10^8$ Kelvin, corresponding to stars on the order of 10 or so solar masses. There's a substantial mass/temperature range between these stars and stars where pair production is important.
This pdf has a couple interesting graphs in the $\rho-T$ plane that you might be interested in. Here are two on radiation pressure and gas pressure:

Here's one on pair production:

(I'll admit that the 100 M$_{\odot}$ track seems suspiciously low, but it's still not yet near the threshold for pair production to become important.)
What happens in stars that have masses high enough for pair production to become truly significant? This limit isn't at 100 solar masses, but a bit higher; Wikipedia states that at 130 solar masses, a pair instability supernova may occur. At this point, you do have conflict between pair production and radiation pressure.
