Solid-state high harmonic generation (sHHG) spectroscopy is a promising technique for studying electronic structure, symmetry, and dynamics in condensed matter systems. Here, we report on the implementation of an advanced sHHG spectrometer based on a vacuum chamber and closed-cycle helium cryostat. Using an in situ temperature probe, it is demonstrated that the sample interaction region retains cryogenic temperature during the application of high-intensity femtosecond laser pulses that generate high harmonics. The presented implementation opens the door for temperature-dependent sHHG measurements down to a few Kelvin, which makes sHHG spectroscopy a new tool for studying phases of matter that emerge at low temperatures, which is particularly interesting for highly correlated materials.