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Classification of documents according to a custom internal hierarchical taxonomy is a common problem for many organizations that deal with textual data. Approaches aimed to address this challenge are, for the vast majority, supervised methods, which have the advantage of producing good results on specific datasets, but the major drawbacks of requiring an entire corpus of annotated documents, and the resulting models are not directly applicable to a different taxonomy. In this paper, we aim to contribute to this important issue, by proposing a method to classify text according to a custom hierarchical taxonomy entirely without the need of labelled data. The idea is to first leverage the semantic information encoded into pre-trained Deep Language Models to assigned a prior relevance score for each label of the taxonomy using zero-shot, and secondly take advantage of the hierarchical structure to reinforce this prior belief. Experiments are conducted on three hierarchically annotated datasets: WebOfScience, DBpedia Extracts and Amazon Product Reviews, which are very diverse in the type of language adopted and have taxonomy depth of two and three levels. We first compare different zero-shot methods, and then we show that our hierarchy-aware approach substantially improves results across every dataset.