My CPU is running slow after a system restore (4x0.8 GHz). Turbo Boost is definitely not working. And how can I fix it? Last weekend, my Macbok Air suffered a heart attack and I ended up doing several hard shutdowns due to an unresponsive system. In the end I had to do a full system restoration.

Everything is working fine now except for one thing: The CPU is slow. Much slower.

CPU related computing is much slower than before, even though I can't tell for sure how much exactly. When running Ubuntu, I monitor the clock speed and see that all four cores are running constantly at 0.8 GHz. The same is true for OS X when using the Intel Power Gadget. It used to be 1.8GHz to 2.6GHz. (Macbook Air 5-2, i5-3427U)



$ sudo modprobe msr ; sudo turbostat
cor CPU    %c0  GHz 
         11.50 0.80 
  0   0  13.34 0.80 
  0   2   7.76 0.80 
  1   1  18.47 0.80 
  1   3   6.44 0.80 


Could this be a software related issue? What should I do then?

I guess you see that a hidden assumption is that the firmware is corrupted. This is why I tried to reinstall the latest firmware for my Mac from https://support.apple.com/kb/HT1237. But when I install that file I get the error: "This software is not supported on your system." Most probably because the latest firmware version is already installed.
Using the Intel power gadget (above) shows your CPU running at low speed.

In normal case that would be fine if there is no load.

I have MBA with 1.8 GHz Intel Core i5 and runs at 0.85 at normal use, how ever it does spikes to 1.8 + under load.

Since the computer processing speed depends on more than one factor (CPU +SSD read/Write+RAM)...

I would say you (your mac) is a candidate for the Apple Hardware test.

UPDATE:

If you have faulty battery (as reported by your hardware test based) then your Mac will automatically limit the CPU speed to about 1/2 (as designed by Apple.)

So fix the battery and you good to go.