Can a black hole have negative temperature?

The temperature of a black hole cannot be negative. The temperature is given by:

$$ T = \frac{\hbar c^3}{8\pi kGM} \tag{1} $$

and obviously this cannot be negative as all the quantities on the right hand side are greater than zero.

However the specific heat is negative and I wonder if this is what you are thinking of. If we differentiate equation (1) with respect to mass we get:

$$ \frac{\mathrm dT}{\mathrm dM} = -\frac{\hbar c^3}{8\pi kGM^2} $$

If the black hole gains some heat dQ then it gains a mass given by $c^2\mathrm dM = \mathrm dQ$, and substituting this into the above equation gives:

$$ \mathrm dT = -\frac{\hbar c^3}{8\pi kGM^2} \frac{\mathrm dQ}{c^2} = -\frac{\hbar c}{8\pi kGM} \frac{\mathrm dQ}{M} $$

If we write:

$$ \mathrm dT = \frac{\mathrm dQ}{MC} $$

where $C$ is the specific heat we get:

$$ C = -\frac{8\pi kGM}{\hbar c} $$

The specific heat is negative because if you add energy to a black hole its temperature decreases and if it loses energy by radiation its temperature increases.