Negative temperatures and negative absolute pressures are both possible in physical systems. Negative temperatures arise in (for example) populations of two-state systems, which have a maximum amount of internal energy they can contain; Negative pressure indicates a system in tension and is rare in fluids but common in solids.
My question is, if I start with a system with positive temperature and positive pressure, and pump energy into it until its temperature becomes negative, will its pressure become negative as well, or will it remain positive? Given a statistical ensemble with negative temperature it should be possible to define the pressure in the usual way, from $p/T = \partial S/ \partial V$, and my question is really about whether I should expect this quantity to be negative or positive.
A naïve argument that it would become negative is as follows: let's suppose that my system has the equation of state of an ideal gas ($pV=NRT$) but that it has a heat capacity that varies with $U$ in such a way as to allow negative temperatures. The only reason I'm making such a weird assumption is that I don't know what the actual equation of state should be – I'm not saying I think negative-temperature systems behave like gases. But if we make this assumption anyway then we can see that if $T$ becomes negative then $p$ must also be negative in order for the equation of state to hold, since $N$, $R$ and $V$ all remain positive.
Obviously no real system is likely to behave as described above. To reiterate, I'm not suggesting that a population of spins in a magnetic field would obey the gas equation. However, I would like to know whether real systems will typically behave in a qualitatively similar way, with the pressure and temperature both changing sign at the same time (and doing so by approaching positive infinity and then switching to negative infinity) or whether the temperature can change from positive to negative while the pressure remains finite and positive.
Best Answer
Negative temparature makes sense in statistical mechanics only if the associated Hamiltonian is bounded from above; otherwise the trace in the definition of the partition function does not exist.
In particular, a gas cannot have negative temperature since its Hamiltonian contains a kinetic energy term, which is not bounded from above. (Edit: Since the momentum operator is unbounded, the kinetic energy is unbounded, hence a Hamiltonian involving a kinetic term is unbounded. Thus the trace of $e^{-\beta H}$ is undefined for negative $\beta$. Thus the temperature must be positive. This is due to a purely mathematical property of the spectrum, independent of which energies can be realized in practice.)
Thus the question about its pressure at negative temperatures (and the argument with the ideal gas law) is moot.
Negative temperature is observed in certain spin systems living inside a crystal, and in systems of vortices in 2-dimensional plasmas. (This temperature is not the temperature of the crystal or plasma, but of the subsystem.) See, e.g., http://www.physics.umd.edu/courses/Phys404/Anlage_Spring11/Ramsey-1956-Thermodynamics%20and%20S.pdf
In these systems, the concept of pressure doesn't make much sense; at least it seems not to be used in this context. Formally, the pressure is of course defined. In a simple example, see Example 9.2.5 in my book
http://lanl.arxiv.org/abs/0810.1019
it changes sign with the temperature.