Calligraphy with Sufi content, architecture of Sufi convents, dervish objects like begging bowls, or any form of art practiced by a Sufi can be categorised under the rubric “Sufi art”. On its own, this term does not denote a particular art form or style.
Sufism encompasses widely different approaches to tasawwuf, which in the widest sense of the word is an approach to Islam that prioritises the quest for divine truth that is assumed to be hidden under the layers of everyday material things and social experiences. Sufis adopt different ways of doing this, usually under the guidance of a spiritual mentor known as shaykh. Because the methods are various, and sometimes contentious from the viewpoint of orthodox Islam, Sufi networks known as tariqas have had various positions in Islamic societies.
The rule of the Mamluks in Egypt and Syria (1250-1517) coincided with a wider phenomenon in the Islamic world that started before the Mamluks and continued after them, namely the institutionalisation and politicisation of Sufi tariqas. Many Sufi shaykhs in this period enjoyed the patronage of ruling elites, and the Mamluks were not an exception. Mamluk sultans and high officers built a number of monumental Sufi convents (khanqahs) as part of their multi-functional charitable complexes, especially in Cairo. The khanqahs accommodated resident dervishes and their rituals, as well as travellers who were either dervishes themselves or enjoyed the company of dervishes. These complexes were supported by endowments that provided for the upkeep of the buildings as well as the needs of the dervishes, so that they could focus on their quest for divine truth without worrying about how to sustain themselves.
It must be noted that given the aim of tasawwuf to affect a dissociation from the material world and its institutions, some Sufis were not happy with these developments. In particular, the need to dissociate from the material world and its institutions did not sit comfortably with the patronage patterns that defined Islamic art and architecture. In the Mamluk period, as elsewhere in the Islamic world, whilst some Sufis rejected patronage, others were well-connected in high places and enjoyed high-status patronage.
Sources where you might find answers to this question:
Leonor Fernandes. The Evolution of a Sufi Institution in Mamluk Egypt: The Khanqah. Berlin: Klaus Schwartz Verlag, 1988.
Ladan Akbarnia & Francesca Leoni. Light of the Sufis: The Mystical Arts of Islam. London and New Haven: Yale University Press, 2010.
Courtney Stewart. ‘Art of the Sufis’. Heilbrunn Timeline of Art History. New York: The Metropolitan Museum of Art, 2000–. http://www.metmuseum.org/toah/hd/sufi/hd_sufi.htm
Doris Behrens-Abouseif. ‘Craftsmen, upstarts and Sufis in the late Mamluk period’. Bulletin of the School of Oriental and African Studies 74 (2011), pp. 375–395.