The incident happened between 10pm on Friday July 19 and 7am on Saturday July 20.
Police are requesting the public to report any suspicious behaviour or witnesses in the area overnight.
A spokesperson from North Yorkshire Police said: "St Catherines Hospice is a charity which helps individuals in their last stages of life suffering terminal illnesses and funds the staff delivering the exceptional help as well as supporting families of those suffering.
"Incidents like this can affect the running of the hospice and impact severely on those in need and therefore anyone involved in incidents like this will be dealt with robustly by North Yorkshire Police."
Anyone who has information which could assist the investigation should contact 101 quoting reference number NYP-20072019-0121.
If you wish to remain anonymous, you can pass information to Crime stoppers on 0800 555 111.