Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function. Add oil. Saute chicken, onion, and garlic until fragrant, 1 to 2 minutes; do not brown garlic.
Add broth, black beans, rice, corn, coconut milk, hot sauce, thyme, black pepper, brown sugar, and allspice and stir well. Close and lock the lid. Select high pressure and close vent. Set timer for 15 minutes. Allow about 10 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 40 minutes. Unlock and remove the lid.
