Why does the spool speed increase with altitude for a turbofan engine (single spool & double spool)? So, I want to see the effect of altitude on the compressor characteristic of a single spool &amp; double spool turbofan engine at constant flight speed and TIT (turbine inlet temp). 

I am aware that as the altitude increases, density decreases, mass flow decreases which ultimately reduces thrust. 

As well as, with an increase in altitude the fuel consumption decreases, due to a decrease in inlet temperature which leads to a lesser amount of compressor work to be done* (From e.g. seventh edition Gas Turbine Theory by H.I.H. Sarvananamuttoo, G.F.C. Rogers, H. Cohen, P.V. Straznicky and A.C. Nix). 

But as the compressor work is directly proportional to the spool speed, I would expect the spool speed to be lower as well. (Correct me if I am wrong). But in the plot shown, the shaft speed parameter (N/$\sqrt(T_{02})$) seems to increase from 0.73 to 0.89 for single spool and 0.86 to 0.91 for double spool with increase in altitude. Why does the spool speed increase with altitude?
As you note - as altitude increases, air density decreases.

The work done by the compressor stages therefore also decreases at any given rotational speed. If we assume constant power applied to the compressor stages (not entirely correctly) then the speed will increase until the work done matches the power input.

The overall work done by the engine will still decrease, due to decreased combustion, but the spool speed is going to increase a certain amount.