I am working with the heat kernel on the hyperbolic space explicitly (as you may guess by my previous questions) and I got the desired results when the curvature is $-\kappa=-1$. Now I am trying to do the same for a fixed but arbitrary curvature $-\kappa<0$, so I need to generalize the explicit formulas for the heat kernel that I got in $\mathbb{H}^n(1)$ (see ''The Heat Kernel on Hyperbolic Space'' or ''Heat kernel bounds on hyperbolic space and Kleinian space'') to a more general $\mathbb{H}^n(\kappa)$.

I achieve to get a general formula when $n$ is odd, that is $$ p_{n}(\rho, t)=\frac{(-1)^{m}}{2^{m} \pi^{m}} \frac{1}{(4 \pi t)^{\frac{1}{2}}}\left(\frac{\kappa}{\sinh (\kappa\rho)} \frac{\partial}{\partial \rho}\right)^{m} e^{-\kappa^2m^{2} t-\frac{\rho^{2}}{4 t}} $$ with $n=2m +1$, but I did it by just adding $\kappa$ in the formula and then cheking that it satisfies the equation, that is now $$\frac{\partial^2}{\partial \rho^2}p_{n}(\rho, t)+(n-1)\kappa \coth(\kappa \rho)\frac{\partial}{\partial \rho}p_{n}(\rho, t)-\frac{\partial}{\partial t}p_{n}(\rho, t).$$ When I try to do the same to even $n$, it fails because I can not compute the derivatives so easily.

Could someone help me find a general formula when $n=2m+2$? I have some intuition but I can not check if they are, in fact, the fundamental solution that I want.