How does Hydrogen gas thermal conductivity vary with temperature. 
What is the theoretical basis of this variation?
Can you point to any applicable references?

Heat transfer in gasses occur with intermoleculer collisions.
Gases thermal conductivity will increase with temperature because
intermoleculer interactions will increase at elevated temperatures
(brownian motion).

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thercond.html

A simple analysis of the heat transported a static ideal gas results
in the following expression for the heat capacity:

  K = n*<c>*l*c_v/3

(see <http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thercond.html>)

where K is the thermal conductivity [J/(cm*s*deg)], 
n is the density of the gas [moles/cm^3],
<c> is the average speed of the gas particles [cm/s],
l is the mean free path of the gas particles [cm], and
c_v is the specific heat capacity at constant volume [J/(mol*deg)]

Several of these quanties depend on temperature.  Specifically, for an
ideal gas, the distribution of molecular speeds is given by the
Maxwell Distribution.  For a given temperature, the average speed of
this distribution is given by:

<c>(T) = (8*R*T/pi*M)^1/2

(see <http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html>

where R is the gas constant {J/(mol*deg)],
T is the absolute temperature [(deg)], and
M is the molar mass of the gas [grams/mol]

For an ideal diatomic gas, there is the well-known result that the
constant-volume heat capacity is given by:

c_v = 5/2*R

(see <http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html#c2>)

Plugging these expressions into the equation for K results in the
following expression for the thermal conductivity as a function of
temperature:

K(T) = 5/3 * n * l * {(2* R^2 * T)/(pi * M)}^1/2

So the thermal conductivity of a given ideal diatomic gas at constant
density (i.e., constant volume and number of gas molecules) should be
proportional to the square root of the absolute temperature.