SAN JOSE, Calif. — There is something that has become ever more apparent in the demeanor and attitude of Henrik Lundqvist over the past two years or so, and it’s only natural.

The Rangers goalie is becoming acutely self-aware.

It becomes even more pronounced at times like this, when the 36-year-old was named to another All-Star team, the fifth of what is surely a Hall of Fame career. He won the only goaltending part of the skills competition Friday night — known as “save streak” — when he made 12 consecutive stops on breakaways, the winning one coming against former Islanders captain John Tavares. And he was then terrific while backstopping his Metropolitan Division team to the All-Star Game victory Saturday night, beating the Central Division, 10-5, in the 3-on-3 final while getting his share of the $1 million purse.

But the difference from years past is that up in the stands were his two daughters, now 6 and 4 years old — “if they were still awake,” Lundqvist said.

As he reflected, Lundqvist also made it clear his kids are now of the age where they are starting to understand what daddy does, and why he is away so much. And that pulls at his heartstrings a little bit as his career segues into the early parts of twilight.

“It’s special to bring your kids,” Lundqvist said before he stopped all six shots he faced in his 10 minutes of the championship game. “I think when you’re younger and you don’t have kids, you don’t realize how much you’re gone, how much you travel, how much you commit to this game. But when you do have kids and they start asking you all the time why you’re leaving, why you’re gone, why you’re not home, then you start to pay attention. ‘Wow, I’m really spending a lot of time [away] and I’m really committed to playing this game.’

“So it’s fun when you can you bring them on the road like this and show them.”

Lundqvist has carved out a terrific life for himself and his family, maybe one that was unexpected after the Rangers made the draft pick of a generation by finding him in the seventh round (No. 205 overall) in 2000. Now, he knows he is a lot closer to the end than the beginning, with two more years left on his current contract with an annual salary-cap hit of $8.5 million.

He has a full no-move clause, so it would be up to him if he decides that he might want to chase that elusive Stanley Cup with another franchise. But as of now, that is not the case. Even though the Rangers are going through a rebuild, and it doesn’t seem like they’re right around the corner from contending for a deep playoff run, Lundqvist remains with the only NHL team he has ever known — and the only American city he has ever called home.

“I take a lot of pride in being a New York Ranger,” Lundqvist said. “It means a lot to me to have this opportunity to play my entire career with one organization. When I came into the league 14 years ago, I didn’t think much about where I was going to be in 10, 15 years. But looking at it now, the way the organization committed to me over the years is a great feeling. I value that a lot.”

The organization also values him a lot, with first-year head coach David Quinn having been effusive in his praise of the netminder. Quinn also noted Lundqvist was their best player and led the charge during a 9-1-1 run just before Thanksgiving. The team has since hit the skids a bit, and Lundqvist has also shown some signs of fatigue after he played 32 of the first 39 games.

But there were still stretches like the two-game road trip just before the New Year, when he carried them to victories in Nashville and St. Louis. Sure, this isn’t one of the Rangers teams from the past whose only measure of success was contending for the Stanley Cup. And it’s not quite the same Lundqvist.

But he knows that, and is OK with it. It’s only natural that perspective grows with age.

“You can’t compare to where we were a couple of years ago, you can’t put that same pressure on,” he said. “I just try to deal with that the best way I can, and really the best way for me is to focus on my game and do what I can to help the team.”