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