A similar question was answered here: Is simply typed lambda calculus equivalent to primitive recursive functions
What I conclude from the answers is that the complexity is that of the extended polynomials, but this refers to what can be computed in the STLC, not of type checking expressions in the STLC.
This link provides an algorithm for type-checking. It is claimed that type checking is linear time in an answer to this question: Complexity of type-checking in relation to complexity of normalization
Also, a survey of algorithms that do type inference for STLC, using unification occurs here, but no complexity result is given.