Temperature of fusion in the Sun vs. fusion in controlled experiments on Earth

The deuterium-tritium fusion reaction cross-section is highly temperature dependent and peaks at temperature of about $8\times 10^{8}$ K, so I suppose these are the temperatures to aim for in a controlled nuclear fusion experiment. In fact according to this, the operating temperatures are at least $10^{8}$ K.

The density of the fusion plasma is a factor - the reaction rate will be proportional to the product of the densities of the two reactants. In fusion reactors the density is of order $10^{20}$ m$^{-3}$. At the centre of the Sun the particle densities are 12 orders of magnitude higher, so partly the increased temperatures in a fusion reactor are to compensate for the lower densities. However, it is also worth remembering that the Sun is not a particularly intense fusion reactor. It only produces about 250 W per cubic metre in its core. A bigger compensatory factor is that the Deuterium-tritium fusion cross-section is about 25 orders orders of magnitude greater than that for proton-proton fusion in the Sun.

In this question I have posted an answer that estimates the energy release per unit volume in typical reactor conditions versus the Sun. I find (order of magnitude) that you get $10^{4}$ times more energy per unit volume out of a reactor than the core of the Sun. So $\sim 10^{6}$ W m$^{-3}$, which I guess is what you will need to make it commercially viable. If you dropped the temperature at all it would rapidly become unviable as a significant power source without absolutely enormous reactors.

This is really just a footnote to Rob's answer.

The Sun is an absolutely terrible fusion reactor. It uses a reaction $p + p \rightarrow d$ that is hopelessly inefficient. The $d + t \rightarrow He + n$ reaction that we use in fusion reactors is (up to) 26 orders of magnitude faster. As Rob says in his answer, the power produced per cubic metre in the Sun is embarrassingly low. However the Sun has the big advantages that its core (where the fusion occurs) is very, very big and very, very dense. The fusion reactors we've managed to make so far are small and the plasma is little different to a vacuum - the particle number density is about one millionth of the density of air.

All this means that our fusion reactors need all the help they can get if they're going to produce a useful amount of power. Adjusting the temperature to maximise the $d + t$ cross section is one of the ways we can boost the power output. The Sun isn't at the optimum temperature for fusion but, well, it's big enough and dense enough that it doesn't care.