I've did web search jmeter and openjdk 13 and found only that relevant post: Is JMeter is going to be paid as Oracle decides for Java post 1.8v?:
JMeter needs Java >=8 (Java 9 and 10 are intermediate and not advised).
On Apache JMeter page last JMeter version mentioned is JMeter 2.5.x (JVM 1.5+).
How does JMeter works on latest OpenJDK 13?
Latest JMeter 5.2.1 works with Java 13
JMeter works for me on a freshly installed Ubuntu 19.04 Desktop with either OpenJDK 13 (installed from the Ubuntu repo) as well as Oracle JDK 13
JMeter officially don't limit java version in requirements:
JMeter is compatible with Java 8 or higher. We highly advise you to install latest minor version of your major version for security and performance reasons.
Also you can see JMeter is tested with Java 13 on Travis CI