You are correct, thermodynamically the total temperature should not vary across a shock, unless it it influenced by the heat transfer from the wall, which is not your case at least in the bow shock away from the wall. It looks like you are using Ansys Fluent. Try to re-start the simulation from the converged solution but use the second order upwind option and then ramp up to the third order muscl scheme for all variables. See whether this reduces the error. The flow is steady (ish as the shear layer downstream of the model may decide to flap around) so there is no requirement for going for an explicit solver. Nevertheless, you can stick with this choice but you may wish to drop the Courant number to 0.8.

One of the most important assumptions in shock wave analysis is that flow across the shock is adiabatic; then, total temperature will be constant through the shock.

There may be real gas effects creeping into your solution which is not exactly captured using a perfect gas assumption. In such a case it is the total enthalpy that remains constant, and the enthalpy of the gas at such elevated temperatures may not be a simple function of just the average translational temperature.

If you are interested in gathering information about shock shapes it is generally a good idea to start with an inviscid solver before moving to viscous solutions.

Q: Do you consider steady or dynamic transient solutions ?

Florian De Vuyst. 

Im considering Steady Solutions.

If I see correctly your figure, I see a narrow green layer that shows first a decreasing then the temperature increases in the red region. It looks more like a numerical oscillation, despite the fact you are using first order upwind.

Are you sure the solution is stopped at an acceptable steady state condition?

First I would check the effects of the turbulence model added in the simulation. I don't think it is necessary in a first step. Then, I would check the sensitivity of the solution to the grid refinement.