Single shot hybrid fs/ps-CARS spectroscopy of N2 is demonstrated at repetition rate up to 5 kHz using an amplified probe delivering a constant energy per pulse between 1 and 5 kHz. We performed 5 kHz CARS thermometry in a laminar CH4/air flame and in ambient air, with a precision under 0.5% at typical flame temperature, which is 2 times more precise and 5 times faster than previous state of the art with this technique. Temperature was measured during long acquisition times up to 100 s, making the system suitable to record signals in the 0.01-2500 Hz spectral window; in our case 10 Hz temperature oscillations were probed.