hot running isn't usually an issue with the Z125.
I have seen and experienced this on my ZG127 which is almost identical.
Kohler. Stupid Kohler. I hate Kohler. But I have a Kohler and am looking at putting a Kawasaki on it.
Anyway....there are two possibilities. One, starter is failing. Not uncommon on Kohler, particularly the SV840's. With that said Kohler updated their replacement starters. They have about 40% more torque, and the main shaft is considerably beefier. It cranks faster, quieter, and lasts. There is also the chinese replacement starter option which I am not a fan of. I've had like 7 or 8 or 9 of them, they never last very long--but in my case, they have warranty so I keep replacing it. I think I have a spare too just in case. What causes the starter to fail? #1 cause is lack of maintenance. Valve adjustment. The Koher needs it's valves adjusted once in a while. When the clearance opens up (which is normal wear characteristic), the compression release fails to release enough compression, the pressure jumps way up, and the starter won't turn it over--a lot of times it will break the old-style sharter shaft completely off. Seen that once on mine, a bunch of times when I worked at the dealer.
Another possibility is the relay is failing. I had to replace a bunch of those too. If you do, the torque spec for the two nuts that hold the cables on, is 4 inch pounds. Snug. not tight just snug. If you overtighten it it will fail again. Guaranteed.
possibly a safety switch issue too. Has it been updated? New style brake switch? Call dealer and ask them if it's been updated, they can look up the serial number. They had a massive repair campaign dealing with it, but it is expired by about 2 years now, so if yours hasn't been updated, you get to buy it and install it...or pay the dealer...and it's pretty involved, they have to replace the crossmember under the fuel tank, which means a lot of disassembly. I did a bunch of them and after while began to hate them!