Context : The Galactic Cold Cores project has carried out Herschel photometric observations of 116 fields where the Planck survey has found signs of cold dust emission . The fields contain sources in different environments and different phases of star formation . Previous studies have revealed variations in their dust submillimetre opacity . Aims : The aim is to measure the value of dust opacity spectral index and to understand its variations spatially and with respect to other parameters , such as temperature , column density , and Galactic location . Methods : The dust opacity spectral index \beta and the dust colour temperature T are derived using Herschel and Planck data . The relation between \beta and T is examined for the whole sample and inside individual fields . Results : Based on IRAS and Planck data , the fields are characterised by a median colour temperature of 16.1 K and a median opacity spectral index of \beta = 1.84 . The values are not correlated with Galactic longitude . We observe a clear T – \beta anti-correlation . In Herschel observations , constrained at lower resolution by Planck data , the variations follow the column density structure and { \beta _ { FIR } } can rise to \sim 2.2 in individual clumps . The highest values are found in starless clumps . The Planck 217 GHz band shows a systematic excess that is not restricted to cold clumps and is thus consistent with a general flattening of the dust emission spectrum at millimetre wavelengths . When fitted separately below and above 700 \mu m , the median spectral index values are { \beta _ { FIR } } \sim 1.91 and { \beta ( mm ) \sim 1.66 } . Conclusions : The spectral index changes as a function of column density and wavelength . The comparison of different data sets and the examination of possible error sources show that our results are robust . However , \beta variations are partly masked by temperature gradients and the changes in the intrinsic grain properties may be even greater .