Consider a classical "degree of freedom" that is linear rather than quadratic: E = c|q| for some constant c

The partition function is:

Z = Sum over all states r of exp[-beta E_{r}]

In the classical limit you can replace the summation by an integration. To do that you need to know how many states there are inside a volume element in a region of momentum and ordinary space. The number of states for a particle inside an n-dimensional volume d^n x and d^n p in momentum space is:

d^n p d^nx/h^n

The partition function can thus be written as:

Z = Integral over x and p of d^nx d^n p/h^n exp[-beta E(x, p)]

If the energy does not depend on position and the particle is confined in a volume V and the x-integration yields a factor of V, where V is the volume:

Z = V Integral over p d^n p/h^n exp[-beta E(p)]

In this case E(P) = c|p|. If we first integrate over all directions in momentum space we are left with an integration over the absolute value of p:

Z = V Integral over |p| from zero to infinity of dp Omega |p|^(n-1)/h^n exp[-beta c |p|]

where Omega p(^n-1) is the area of an n-dimensional sphere

Substitute x = beta c |p| in this integral to obtain

Z = beta^(-n) *const.

where const does not depend on beta.

The average energy is given by the derivative:

E = -d log[Z]/ d beta = n/beta = n kT

In one dimension this is indeed kT. In general this is twice the average energy for a nonrelativistic particle.