What would a blackbody sound like?
This problem can be solved with noise-shaping. Since the shape of the spectrum is known, it can be used as a base for the power spectral density:
$$ P(f,T)=\frac{ 2 h f^3}{c^2} \frac{1}{e^\frac{h f}{k_\mathrm{B}T} - 1} $$
where $k_\mathrm{B}$ is the Boltzmann constant, $h$ is the Planck constant, and $c$ is the speed of light. This outputs the relative power of each band as a continuous function of frequency, $f$, and temperature, $T$. Since the output quantity must be expressed in decibels (dBr) to be meaningful for audio, we simply use a log scale and add an offset (a gain) to normalize the peak to 0. The equation of the EQ curve is:
$ E(f,T) = 10 \log{ P(f,T) } + G_{t}(T) $
where $G_{t}(T)$ is the gain required to normalize the peak to 0 dB. The required gain depends on the inverse cube of the temperature plus a constant, $G$ (187 dB); thus, $ G_{t}(T) = G - 10 \log T^3 $. The leading coefficient $10$ converts bels to decibels. Simplifying gives us:
$$ E(f,T) = 10 \log{ \left( \frac{ 2 h f^3}{c^2 T^3} \frac{1}{e^\frac{h f}{k_\mathrm{B}T} - 1} \right)} + G $$
tl;dr:
We obtain our waveform by applying an EQ to gaussian white noise from AudioCheck.net.
Examples:
17 nanokelvins is the temperature at which black noise has a peak frequency of 1 KHz. Its passband is limited to 1 Hz to 12 KHz.
30 nanokelvins is the lowest temperature at which black noise has a passband that spans the entire hearing range.
55 nanokelvins is the temperature at which black noise has a peak frequency of approximately 3 KHz, the most sensitve frequency of human ears.
340 nanokelvins is the temperature at which black noise has a peak frequency of just under 20 KHz, which is the limit of human hearing. Most of the audible spectrum is a linear upward ramp, which is very similar to violet noise. At higher temperatures, the frequency domain will be almost identical to violet noise.
All EQ filter parameters are in the descriptions of the tracks on SoundCloud.
If you are cooling your object that you wish to hear, then the exact sound will depend on the exact temperature (as given by yuki96's answer at 17nK).
However, any temperature above the nanoKelvin temperature scale will sound the same, but the volume will increase with temperature (according to the Stefan-Boltzmann law).
The sound of a warm blackbody (such as what you would get at room temperature) would sound like a violet or purple noise. You can listen to a sample of purple noise here.