1. The cost for the maintenance of the house has gone up recently.
2. The cost for the maintenance of the house has just gone up.
(Q1: What is the use/case of the present perfect in this sentence? This sentence contains 'recently.' Does the sentence tell 'the finished activity' of 'experience'? Q2: Are both the same in meaning?)
The preposition after "cost" needs to be "of" not "for."
Any time the present perfect is used, it means that the verb's action took place in the past and continues into the present.
Both sentences mean essentially the same thing.