Xapain builds under 3.13 are broken. The version of the xapian lib is 1.4.18,
so I am pulling the same version of the source code so it's compatible.
I can't use apt to install the python xapian extension because it puts it
in the OS's python, not in the python I am using to test.

The error is:

...
make  all-am
make[3]: Entering directory '/tmp/xapian-bindings-1.4.18/python3'
depbase=`echo xapian_wrap.lo | sed 's|[^/]*$|.deps/&|;s|\.lo$||'`;\
/bin/bash ../libtool  --tag=CXX   --mode=compile g++ -DHAVE_CONFIG_H -I. -I..  -
I/opt/hostedtoolcache/Python/3.13.0-alpha.6/x64/include/python3.13  -fno-strict-aliasing -Wall 
-Wno-unused -Wno-uninitialized -fvisibility=hidden  -g -O2 -MT xapian_wrap.lo -MD -MP -MF 
$depbase.Tpo -c -o xapian_wrap.lo xapian_wrap.cc &&\
mv -f $depbase.Tpo $depbase.Plo
libtool: compile:  g++ -DHAVE_CONFIG_H -I. -I.. -I/opt/hostedtoolcache/Python/3.13.0-
alpha.6/x64/include/python3.13 -fno-strict-aliasing -Wall -Wno-unused -Wno-uninitialized -
fvisibility=hidden -g -O2 -MT xapian_wrap.lo -MD -MP -MF .deps/xapian_wrap.Tpo -c 
xapian_wrap.cc  -fPIC -DPIC -o .libs/xapian_wrap.o
xapian_wrap.cc: In constructor 
'XapianSWIG_Python_Thread_Block::XapianSWIG_Python_Thread_Block()':
xapian_wrap.cc:87:13: error: 'PyEval_ThreadsInitialized' was not declared in this scope
   87 |         if (PyEval_ThreadsInitialized()) {
      |             ^~~~~~~~~~~~~~~~~~~~~~~~~
xapian_wrap.cc: In constructor 
'XapianSWIG_Python_Thread_Allow::XapianSWIG_Python_Thread_Allow()':
xapian_wrap.cc:109:47: error: 'PyEval_ThreadsInitialized' was not declared in this scope
  109 |     XapianSWIG_Python_Thread_Allow() : status(PyEval_ThreadsInitialized()) {
      |                                               ^~~~~~~~~~~~~~~~~~~~~~~~~
make[3]: *** [Makefile:782: xapian_wrap.lo] Error 1
make[3]: Leaving directory '/tmp/xapian-bindings-1.4.18/python3'
make[2]: *** [Makefile:682: all] Error 2
make[2]: Leaving directory '/tmp/xapian-bindings-1.4.18/python3'
make[1]: *** [Makefile:521: all-recursive] Error 1
make[1]: Leaving directory '/tmp/xapian-bindings-1.4.18'
make: *** [Makefile:443: all] Error 2

xapian is disabled for now on GitHub 3.13 CI testing.

Consider forcing use of 1.4.25 (released 2024-03-07)
https://oligarchy.co.uk/xapian/1.4.25/xapian-bindings-1.4.25.tar.xz
rather than same version of installed library. IIUC all 1.4 should be compatible:
  https://trac.xapian.org/ticket/818 comment 3.

Same error with 3.13beta1 and 1.4.18.

If I force use of 1.4.25, I get a different error (along with a warning of mismatching
1.418 and 1.4.25 versions):

configure: WARNING: Xapian library is version 1.4.18 but the bindings are version 1.4.25 - we 
strongly recommend using matching versions.
case "$PYTHON_VERSION" in "3.14") echo skipping xapian build;; *) make && sudo make install; 
esac
+ case "$PYTHON_VERSION" in
+ make
make  all-recursive
make[1]: Entering directory '/tmp/xapian-bindings-1.4.25'
Making all in .
make[2]: Entering directory '/tmp/xapian-bindings-1.4.25'
make[2]: Leaving directory '/tmp/xapian-bindings-1.4.25'
Making all in python3
make[2]: Entering directory '/tmp/xapian-bindings-1.4.25/python3'
make  all-am
make[3]: Entering directory '/tmp/xapian-bindings-1.4.25/python3'
depbase=`echo xapian_wrap.lo | sed 's|[^/]*$|.deps/&|;s|\.lo$||'`;\
/bin/bash ../libtool  --tag=CXX   --mode=compile g++ -DHAVE_CONFIG_H -I. -I..  -
I/opt/hostedtoolcache/Python/3.13.0-beta.1/x64/include/python3.13  -fno-strict-aliasing -Wall 
-
Wno-unused -Wno-uninitialized -fvisibility=hidden -fvisibility-inlines-hidden  -g -O2 -MT 
xapian_wrap.lo -MD -MP -MF $depbase.Tpo -c -o xapian_wrap.lo xapian_wrap.cc &&\
mv -f $depbase.Tpo $depbase.Plo
libtool: compile:  g++ -DHAVE_CONFIG_H -I. -I.. -I/opt/hostedtoolcache/Python/3.13.0-
beta.1/x64/include/python3.13 -fno-strict-aliasing -Wall -Wno-unused -Wno-uninitialized -
fvisibility=hidden -fvisibility-inlines-hidden -g -O2 -MT xapian_wrap.lo -MD -MP -MF 
.deps/xapian_wrap.Tpo -c xapian_wrap.cc  -fPIC -DPIC -o .libs/xapian_wrap.o
xapian_wrap.cc: In function 'PyObject* PyInit__xapian()':
xapian_wrap.cc:66318:114: error: 'FLAG_NGRAMS' is not a member of 'Xapian::TermGenerator'
66318 |   SWIG_Python_SetConstant(d, "TermGenerator_FLAG_NGRAMS",SWIG_From_int(static_cast< 
int 
>(Xapian::TermGenerator::FLAG_NGRAMS)));
      |                                                                                                                  
^~~~~~~~~~~
xapian_wrap.cc:66331:99: error: 'SNIPPET_NGRAMS' is not a member of 'Xapian::MSet'
66331 |   SWIG_Python_SetConstant(d, "MSet_SNIPPET_NGRAMS",SWIG_From_int(static_cast< int >
(Xapian::MSet::SNIPPET_NGRAMS)));
      |                                                                                                   
^~~~~~~~~~~~~~
xapian_wrap.cc:66352:110: error: 'FLAG_NGRAMS' is not a member of 'Xapian::QueryParser'
66352 |   SWIG_Python_SetConstant(d, "QueryParser_FLAG_NGRAMS",SWIG_From_int(static_cast< int 
>
(Xapian::QueryParser::FLAG_NGRAMS)));
      |                                                                                                              
^~~~~~~~~~~
xapian_wrap.cc:66355:116: error: 'FLAG_NO_POSITIONS' is not a member of 'Xapian::QueryParser'
66355 |   SWIG_Python_SetConstant(d, 
"QueryParser_FLAG_NO_POSITIONS",SWIG_From_int(static_cast< 
int >(Xapian::QueryParser::FLAG_NO_POSITIONS)));
      |                                                                                                                    
^~~~~~~~~~~~~~~~~

Created a new workflow just for testing xapian build with various libraries.

opened upstream ticket. https://trac.xapian.org/ticket/828

Just tried with 24.04 ubuntu release. installed xapian version 1.4.22. It passes.

Cleaned up special casing of python 3.13 from the workflow job code.

Install xapian job is still disabled for 3.13 since ubuntu-latest is still
22.04. When that updates in a month or so, I'll re-enable the install for python 3.13.