What is the long term fate of the gas giants? If I'm not mistaken, it is believed that the reason for such turbulent weather on the 4 outer gas giant planets is that the internal pressure is so great that it is generating heat, which is causing convection, which causes extreme weather.

Will these planets forever generate heat, or at some point will they freeze up?  

What will Jupiter look like a trillion years from now?
The timescale on which Jupiter cools is reasonably well understood and predicted by the current generation of evolutionary models.

Jupiter's luminosity is provided mostly by gravitational contraction. For a planet that only contains gas governed by the perfect gas law, the appropriate timescale for this contraction (or indeed for the luminosity to fall significantly) is given by the Kelvin Helmholtz timescale.
$$ \tau = \eta \frac{GM^2}{RL},$$
where $M$ and $R$ are Jupiter's mass and radius and $L$ is its current power output (or luminosity), and the parameter $\eta \sim 1$. This timescale is a few $10^{11}$ years. 

However, giant planets like Jupiter are not governed by perfect gas laws. The gas in the centre of Jupiter is dense enough that electrons become degenerate. Degenerate electrons fill the available energy levels up to the Fermi energy. Their consequent non-zero momenta of the electrons exerts a degeneracy pressure that is independent of temperature. As a result, the rate of contraction slows and the release of gravitational potential energy slows; the planet is able to cool and remain in hydrostatic equilibrium without the same degree of contraction.

One can express this change using the $\eta$ parameter. For Jupiter $\eta \simeq 0.03$ (Guillot &amp; Gautier 2014) - i.e. the timescale for the luminosity to fade is 30 times quicker than the naive Kelvin-Helmholtz time and Jupiter's luminosity will scale as the reciprocal of its age and will fall by a factor of a few in $10^{10}$ years. In a trillion years, the luminosity of Jupiter will be lower than it is now by roughly a factor of 250.