Hi Aaron,
in our code, the cs^2 is defined depending on the DESCRIPTOR, so in most cases this is 1/3.
Hence, the pressure should be p = \rho * cs^2.
When you take a closer look at the heat conductivity lambda, it is connected to the thermal diffusivity alpha via alpha = lambda/(\rho*C_p). In your first equation, there is no \rho to be seen since this is all in lattice units where we choose \rho = 1 for simplicity.
Hope this helps.
Kind regards
Tim