  Nonlinear diffusion $\partial_t \rho = \Delta(\Phi(\rho))$ is considered for
a class of nonlinearities $\Phi$. It is shown that for suitable choices of
$\Phi$, an associated Lyapunov functional can be interpreted as thermodynamics
entropy. This information is used to derive an associated metric, here called
thermodynamic metric. The analysis is confined to nonlinear diffusion
obtainable as hydrodynamic limit of a zero range process. The thermodynamic
setting is linked to a large deviation principle for the underlying zero range
process and the corresponding equation of fluctuating hydrodynamics. For the
latter connections, the thermodynamic metric plays a central role.
