I have seen in atleast three places saying that he has written a Shivashtakam (apart from the one by Sri Adi Shankaracharya).
1. backtogodhead (cites the source "Recorded in Murari Gupta’s Sri Caitanya Carita Mahakavya")
But Srila Prabhupada, in Teachings of Lord Chaitanya, says -
"Although Lord Caitanya was widely renowned as a scholar in His youth, He left only eight verses, called Śikṣāṣṭaka.”
This Shivashtakam is different from the popular one which starts from "Prabhum Prananaatham..." by Adi Shankaracharya.
EDIT : I have found the Murari Gupta's Sri Chaitanya Carita Mahakavya here - Where can I get the complete English translations of the Sri Krishna Chaitanya Charitamritam & Sri Chaitanya Charitamritam Mahakavyam? . The link is http://www.krishnapath.org/Library/Goswami-books/Murari-Gupta/Murari_Gupta_Sri_Krsna_Caitanya_Carita_Maha_Kavya.pdf . If someone could locate the Shivashtakam here, as cited in backtogodhead.in, it would be a great favour.