We study a variant of the uncapacitated facility location problem (UFL), where connection costs of clients are defined by (client specific) concave nondecreasing functions of the connection distance in the underlying metric. A special case capturing the complexity of this variant is the setting called facility location with penalties where clients may either connect to a facility or pay a (client specific) penalty. We show that the best known approximation algorithms for UFL may be adapted to the concave connection cost setting. The key technical contribution is an argument that the JMS algorithm for UFL may be adapted to provide the same approximation guarantee for the more general concave connection cost variant. We also study the star inventory routing with facility location (SIRPFL) problem that was recently introduced by Jiao and Ravi, which asks to jointly optimize the task of clustering of demand points with the later serving of requests within created clusters. We show that the problem may be reduced to the concave connection cost facility location and substantially improve the approximation ratio for all three variants of SIRPFL.