For　a　type　of　nonminimally　coupled　vector-tensor　theories　with　Abelian　gauge　symmetry　breaking　in　four-dimensional　spacetime　and　correspondingly　asymptotic　non-AdS　black　hole　solutions　including　a　cosmological　constant,　we　construct　the　appropriate　boundary　terms　and　derive　the　associated　junction　condition.　In　order　to　remove　the　divergences　in　the　stress　tensor　which　is　localized　on　the　spacetime　boundary,　we　also　involve　the　suitable　surface　counterterms　into　the　total　action.　Using　the　counterterm　method,　we　calculate　the　black　hole　mass.　An　implicit　relation　between　the　black　hole　carge　Q　and　other　parameters　is　implied　by　combining　the　expression　of　the　black　hole　mass　with　the　first　law　of　black　hole　thermodynamics.　With　this　implicit　relation,　we　can　prove　the　inequality　Q　＜=　M　which　is　a　general　bound　for　most　of　charged　black　holes.　Besides,　the　phase　structure　of　black　holes　is　also　investigated　in　the　grand　canonical　ensemble.