thermodynamics - Could temperature have been defined as $-partial S/partial U$?

When coming up with a definition of temperature, it's typical to start with an empirical definition that a system with a hotter temperature tends to lose heat to a system with a colder temperature. Combined with the second law of thermodynamics, that leads to the condition that for two systems in thermodynamic equilibrium,

$$\frac{\partial S_1}{\partial U_1} = \frac{\partial S_2}{\partial U_2}$$

(consider it implicit throughout this question that volumes are constant). So temperature has to be some function of $\partial S/\partial U$. Furthermore, we'd like the object that loses the heat to be the one with the hotter temperature, so temperature has to be inversely related to $\partial S/\partial U$.

Of course the conventional choice is $\frac{1}{T} = \frac{\partial S}{\partial U}$, but if we were rederiving thermodynamics and statistical mechanics from scratch, is there any reason we couldn't have chosen

$$B = -\frac{\partial S}{\partial U}$$

instead? Sure, it would lead to many common temperatures being negative, but suppose we're willing to accept that. Is there any other useful property that the standard definition of temperature has which wouldn't be shared by $B$? Would it necessarily lead to nonlinear thermometers, for example?

As Michael Brown pointed out in a comment, this alternative definition is just $B = -k\beta$, so I guess another roughly equivalent way of saying what I'm asking would be, is there a practical reason not to use $\beta$ or $B$ as temperature instead of $T$?

I've already looked at this question and this one and some others, but any close-to-relevant answers there seem to be saying merely that it's conventional to define temperature the way it is defined, not that we couldn't have done it differently.

Answer

If one takes $U$ as the dependent variable and $N, V,$ and $S$ as the independent variables, then one has $U = U(S,V,N)$ with the total derivative of U equal to $$\rm dU = (\partial U/\partial S)_{V,N} \; \rm dS + (\partial U/\partial V)_{S,N} \;\rm dV + (\partial U/\partial N)_{S,V}\;\rm dN.$$ Each of these partial derivatives has a "simple form", with $T = \partial U/\partial S$, $\,-p = \partial U / \partial V$ and $\mu = \partial U/\partial N$. The condition for equilibrium between two systems open to thermal transfer is that the two $T$'s be equal, the condition for two systems open to pressure change is that the two $p$'s be equal, and the same for exchange of particles with the two chemical potentials $\mu$ be the same.

These intensive variables are fundamental for thermodynamics, but could have been defined in any reasonable way, such as $1/T$ ($1/k T$ might have been better) or whatever. So the short of it is that while the derivatives are all important, their actual definition is somewhat arbitrary.