Sterilisation apparatus (1) for sterilising a container (100) comprising: a treatment chamber (8) for treating the container (100) a vaporisation device (2) receiving air and a watery composition containing liquid hydrogen peroxide configured to vaporise and dispense a mixture of air, water vapour and evaporated sterilising agent towards the treatment chamber (8) a post-heating unit (7) operatively active on the mixture so as to maintain the temperature thereof above the dew point temperature, the post-heating unit (7) being located immediately upstream of the treatment chamber (8).